Fortschritte der Physik / Progress of Physics: Volume 32, Number 8 [Reprint 2022 ed.] 9783112656204

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FORTSCHRITTE DER PHYSIK PROGRESS OF PHYSICS

Volume 32 1984 Number 8 Board of Editors F. Kaschluhn A. Lösche R. Rompe

Editor-in-Chief F. Kaschluhn

Advisory Board

A. M. Baldin, Dubna J . Fischer, Prague G. Höhler, Karlsruhe K. Lanius, Berlin J . Lopuszanski, Wroclaw A. Salam, Trieste D. V. Shirkov, Dubna A. N. Tavkhelidze, Moscow I. Todorov, Sofia J . Zinn-Justin, Saclay

CONTENTS: B . N . KALINKIN

The Process of Multi-Hadron Production and the Problem of Colour Confinement

395—448

W . DRECHSLER

Poincaré Gauge Theory, Gravitation and Transformation of Matter Fields

4 4 9 - 472

AKADEMIE-VERLAG • BERLIN ISSN 0016 - 8208

Fortechr. Phys., Berlin 82 (1984) 8, 395-472

EVP 1 0 , - M

Instructions to Authors 1. Only papers not published and not submitted for publication elsewhere will be accepted. 2. Manuscripts should be submitted in English, with an abstract in English. Two copies are desired. 3. Manuscripts should be no less than 30 and preferably no more than about 100 pages in length. 4. All manuscripts should be typewritten on one side only, double-spaced and with a margin 4 cm wide. Manuscript sheets should be numerated consecutively from "1" onwards. Footnotes should be avoided. 5. The titel of the paper should be followed by the authors name (with first name abbreviated), by the institution and its address from which the manuscript originates. 6. Figures and tables should be restricted to the minimum needed to clarify the text. They should be numbered consecutively and must be referred' too in the text and oh the margin. Figures and tables should be added td the manuscript on separate, consecutively numerated sheets. The tables should have a headline. Legends of figures should be submitted on a separate sheet. All figures should bear the author's name and number of figure overleaf. Photographs for half-tone reproduction should be in the form of highly glazed prints. Line drawing should be in a form suitable for reproduction. The lettering should be sufficiently large and bold to permit reduction. If requested, original drawing and photographs will be returned to the author upon publication of the paper. 7. Formulae should not be written to small and not with pencil. Separate lines for formulae are desirable. Si-units should be used. Letters in formulae are normally printed in italics, numbers in ordinary upright typeface. Underlining to denote special typefaces should be done in accordance with the following code: Italics: wavy underlined with pencil (only necessary for type written symbols in the text) Boldface italics (vektors): straight forward and wavy underlined Upright letters (all abbreviations like all units (cm, g, ...), all elements and particles (H, He, ..., n, p, ...), elementary mathematical functions like Re, Im, sin, cos, exp, ...): black underlined Gree*k letters: red underlined Boldface Greek letters: red interlined twice Upright Greek letters (symbols of elementary particles): red and black underlined Large letters: underlined with pencil twice Small letters: overlined with pencil twice (This will be necessary for handwritten letters that do not differ in shape, as c C, k K, o 0, p P, s S, u U, v, V, w, W,xX,yY,z Z). It will help the printer if position of subscripts and superscripts is marked with pencil in the following way: at, 6j, Jftj, Mtj, W Please differentiate between following symbols: a, a; a, a , oo; a, d; c, C, c ; e, I, S, e, k, K, x, x; x, X, x, X ; I, 1; o, 0 a, 0; p, Q; W, U, U. V, V, V, 3GeV.

(2.8)

From the arguments used in deriving this estimation it follows the mass m co „ may slowly grow with energy, since E„° ~ T. I t is interesting to note that the value of m corr (2.8) is in a good agreement with the value, obtained from the analysis of correlation experiments at ISR energies, mcaTr = 4 GeV, in which this characteristics was considered as a free parameter [27]. The boundedness of m corr leads to a number of qualitative consequences for the multiple-production process. Here we cite some of them. For M m corr the process of hadronization should be local as the number of quasiindependent origins of hadronization m = M/mt0„ 1. Manifestations of this property are diverse and important. For instance: Constant time of hadronization rh of objects of the cluster type [24] (the number of hadronization regions grows with energy, while the hadronization time of each region varies slowly in the first approximation); Formation of jets from particles both with small and large ; Local compensation of the charge and transverse momentum; Limitedness of masses of the clusters or the so-called H-quanta [30]. The presence of the limited correlation mass in the hadronization process allows us to get over the difficulties which have been met for a long time in analysing multiple processes within multiperipheral and cluster models (see, e.g., [31]). On the one hand, a number of general properties of the process, like the multiplicity, mean transverse momentum, qualitative composition of produced particles is well described within thermodynamic models which assume the production of one heavy cluster with a single temperature and decay volume. On the other hand, from the correlation analysis data it follows that there exist several comparatively light clusters decaying in the first approximation independently of each other. However, from the positions we have presented there is no contradiction between these facts, as these are manifestations of a unique picture of the decay of an essentially relativistic system obeying the requirement of "colour" nonobservability. Thus, the principle of colour confinement appears to be important not only for the CS global development but also for correlation effects in the hadronization process.

406

B.

N.

KALIKKIN,

The Process of Multi-Hadron Production

Internal collective motion in CS Considering above the production mechanism of a CS in the e+e~-annihilation, for instance, we noted that the field excitation may take away a considerable part of the energy of relative motion in a initial quark-antiquark pair. Being a consequence of dissipation caused by forces of dynamic confinement, this energy part in the thermodynamic language may be considered as the thermal energy. The remaining part of the CS energy appears as the longitudinal collective motion in the system represented as the relative motion of subsystems of the CS. (As we have seen in the preceding section, the introduction of subsystems in a relativistic expanding object like CS is not only justified but also is inevitable). Since at earlier stages of the CS evolution these subsystems are colour objects while the forces of confinement acting between them according to (2.3) depend upon the difference of their rapidities, then if one assumes the absence of discontinuity in matter, a homogeneous distribution of subsystems over rapidity y0 represents an equilibrium configuration. Therefore, the distribution function of subsystems over rapidity will be taken as a table-shaped distribution fiVo =

— ym < ?/« < ym , , - yo - ym lyol >Vm-

i const; n

10;

2.9)

In this case the part of energy transforming into the thermal excitation of a system is

Vm f KVa) zlut vr

w

X — 6 t ~ M ~

fyo

J fiVo) ch

~Vm

y™

yady0

o in\ '(2"10)

Correspondingly, 6C — 1 — dT is the part of energy of the system for the collective motion. Note that a distribution (2.9) is used also in simple parton schemes of hadron interactions. However, its nature is there different. It arises as a result of the equilibrium spectrum of partons before the collision in accelerated cold hadrons. On the contrary, in the model under consideration the form (2.9) is a consequence of the interaction dynamics of coloured objects. Therefore, it is independent of the nature of particles in collisions of which a compound system is formed, and it is universal for all types of interactions accompanied by the multiparticle productions: Hadron interactions, deep inelastic scattering or e+e~-annihilations into hadrons. The model [17] and parton scheme differ also in the definition of the energy dependence of the quantity y m . In the parton scheme

yJ>~\nM/fi

(2.11)

where M is the mass of a compound system,,« is the mass of a produced particle. In the model [17] from comparison of (2.5) and (2.10) it follows that

shymlym~\nM.

(2.12)

I t is seen that in first approximation this dependence is log-logarithmic. For its description in refs. [26, 28, 32] the expression

ym = In (1 +

Cl

In (M/c,))

(2.13)

407

Fortschr. Phys. 32 (1984) 8

was used, in which the constants cx = 4.9 and c2 = 2 GeV are defined from the comparison with experiment. To complete this section we stress that in considering basic properties of CS we used rather general considerations and modern ideas on the structure of matter. In fact we were always based on the quark-gluon structure of hadrons. The principle of "colour" confinement serves as the main factor regularizing both the global and local properties of CS from the moment of its formation to decay. The essentially relativistic nature of CS was taken into account, and the requirement for the causality principle to be fulfilled in these conditions may lead to important consequences for the nature of phase transition quarks + gluons into hadrons. An important fact should be noted that results of the latest studies of the behaviour of Yang-Mills systems may give evidence in favour of the use of statistical-thermodynamical methods for the description of the CS evolution in the conditions of a considerable energy release. And vice versa, they make doubtful the application of models of the QCD type in these cases. III.

Thermodynamical Approach and Some General Characteristics of Multiple Production

First of all we note that: i) a CS, as a rule, is a highly excited object in the continuous spectrum; ii) from the view-point of the quark-gluon picture, hadrons emitted by a CS and "jets" are in turn complicated objects. In accordance, as the first approximation for the inclusive spectrum of particles produced by a CS in refs. [17, 26, 28, 32] the Gibbs thermodynamical distribution was used. Allowing for the collective motion in a CS (2.9) the inclusive spectrum was obtained in the form (in terms of variables p±, y): Vm

-Vm

where m±i = ]/pj_2 + m,2, wij, gt are respectively the mass and statistical weight of a particle of sort " i " , Vh c± 56 fm3 is the average volume of hadronization. Temperature T is defined from the condition of equality of the sum of energies of all produced particles and the total energy of the system M : £ f Ei(dni/dpx dy) dp± dy = M. (3.2) i

We shall employ the relations (3.1), (3.2) and their consequences for analysing the experimental situation. 1.

Characteristics defined by the total mass of a CS. Mean multiplicity

With the use of (3.1) the mean multiplicity for different processes can be written in the form dN-

408

B. N. K a l i n k i s t , The Process of Multi-Hadron Production

Here x; is the number of charged particles produced in the decay of an i-th particle (if it is unstable and its decay is caused by the strong interaction; to mark this possibility, we use the notation Ni instead of nl in (3.3)), rj is the number of leading charged particles produced out of the CS decay. From (3.1) —(3.2) it follows that both the spectra and multiplicity in the first approximation should be the same at equal masses of an intermediate hadron CS, i.e. they should not depend on a specific process where the CS has been formed. Consider now examples. In inelastic scattering of hadrons only a half of energy is on the average spent for the formation of a CS. I n this case the CS mass equals M = k^s, where k pa 0.5 is the inelasticity coefficient. As is known, in this case rj ~ 1.5. In accordance with the quarkgluon picture, leading particles produced in this process result from the hadronization of a jumping forward colourless weakly excited system of valence quarks. The CS at the first moment is a result of stripping and merging of gluon fields of reactive hadrons_

/s/0eV Fig. 2. Multiplicity of charged particles as a function of / s : the curve "1" represents an averaged act of multiple production (inelasticity coefficient k = 0.5), curve "2" represents the cases of formation of a CCS (h = 1).

Then there occurs an intensive production of quark-antiquark pairs, and a further evolution of the CS goes in accordance with the above described recipe. Calculated dependences of ( n ± ) on ]/s for pp-interactions are compared with experimental data (see review [33] and paper [34]) are given in Fig. 2 b y curve "1". Curve,"2" is calculated at k = 1, rj = 0. These values correspond to the formation of the complete compound system (CCS) which possesses the whole energy of colliding hadrons. Only such systems are formed in the process of e + e~-annihilation into hadrons since the leading effect peculiar to hadron interactions is absent here (there is no singlet of valence quarks made a priori). From Fig. 2 it is seen that curve " 2 " satisfactorily describes the (n±) dependence on jIs for e+e~-annihilation into hadrons [25]. The describing the case when all the CS energy is thermal (not shown in the figure) would be by the l.h.s. of curve "2", nearly in the same distance as curve "1". I t is seen that the inclusion of collective motion is important. The complete compound systems with cross section a c are formed also in hadron collisions. J u s t this channel leads, for instance, to the production of particles with large

409

Fortschr. Phys. 32 (1984) 8

(see chap. V). Indeed, the data on the magnitude of ( n ± ) in events where there is a hadron with > 3 GeV/c [36] are situated on curve "2". In deep inelastic scattering of a lepton on a valence quark of the target nucleon the leading hadron is also absent. Therefore, in this case rj = 0 while the mass of the hadron system M is defined experimentally. The result for the (n±) dependence on M2 for vp-interactions and experimental data [37] are drawn in Fig. 3.

+

/ f 100 U00

U 610 20 s/6eV.2

Fig. 3. Behaviour of (» ± ) in the case of vp-interaction

Mean transverse momentum of particles Long ago it has been established that this characteristic in thermodynamical model is the most simple and depends only on the temperature T of the CS decay (for 71-mesons ~ T). Since the particle spectrum is the same for CS with the same mass the (•p_,_> should be the same for them. Insignificant deviations may occur because of a somewhat different contribution of particles produced at earlier stages of the CS evolution (in different processes the CS initial volumes may be different). The dependence of (pj_) on ]/s (the result of averaging over the distribution (3.1)) for the process of e + e~-annihilation into hadrons and inelastic pp-interaction is shown in Figs. 4, 5. As is seen, the calculation is in satisfactory agreement with experiment.

0M

^T 0.3

J*

Fig. 4. Function ( p ± ) = f(fi) for e + e~-annihilation into hadrons

'

10

20

50

ST/GeV

0.6

0ti

Fig. 5. Function ( p = f(l/s) in an averaged act of multiple production in pp-interaction (read (p ± ) instead of p L at the ordinate)

02 10

10J

102 n

E'p /Gev

410

B.

N.

The Process of Multi-Hadron Production

KALINKIN,

Qualitative composition of produced particles An attractive feature of the thermodynamic approach is the possibility to avoid the arbitrariness in the definition of qualitative composition of produced particles. Admitting temperature variations of the decay T to obtain the observed value of and a change of Vh as to reproduce (n±), we get a scheme of the qualitative composition without free parameters. However, despite such a rigid condition the thermodynamical model satisfactorily describes the qualitative composition of produced particles. In Figs. 6 and 7 we draw the results of calculations and experimental data [38, 39] on yields of Ri-mesons and antiprotons for pp-collisions and e+e~-annihilation into hadrons.

102

103

Fig. 6. Yield of K ± -mesons and antiprotons in an averaged act of pp-interaction

s / Gev2

0.6

c

0.2

Fig. 7. Yield of K ± -mesons and antiprotons in e+e~-annihilation into hadrons

0.06

0.02

3.5

40

4.5 f s / OeV

5.0

Anisotropy of the CS decay The degree of anisotropy of the CS decay and its "jet" structure is described with the help of certain characteristics. The use is frequently made of the spherocity

S' =

min

^

^

t i

E\Vi

(3.4)

and trust Tr =

max

Z bn.il I w '

(3.5)

where the longitudinal p\\ and transverse momenta of particles are defined with respect to an axis which minimizes S' or gives a maximal Tr (for the trust we use Tr instead of conventional T that in our paper stands for temperature).

411

Fortschr. Phys. 32 (1984) 8

These characteristics are closely related to the collective motion in a CS. For instance, in the model [17] the spherocity S' is directly expressed via the fraction of the thermal energy in a CS. Using (3.1) we get

Comparison of (3.4) and (3.6) gives S' = V -

(3.7)

The values of (S') and 1 — Tr calculated as functions of ]/s within the model [17] and experimental values for e+e~-annihilation into hadrons [40] are presented in Fig. 8. o.u 0.2 0.1

Fig. 8. Behaviour of the "spherocity" S' and trust T in e+e--annihilation into hadrons 10

20 SO jt+ + X. Curve " 2 " is the calculation without the consideration of collective motion 6

) If condition « = const (s) (see eq. (4.4)) is fulfilled.

425

Fortschr. Phys. 32 (1984) 8

In fig. 22 the consequences of the model are compared with the results of measurements of the inclusive spectra of positive particles at different values of the Feynman variable x [53].

Fig. 22. Dependence of (E(d 3 ajdp 3 )) on uJf0-6U

3.

. i V 2 3U p± / (GeV/c)

„

different x = 2pitl^s

for

at

= 52 GeV

On the production of jets with large Pj_

The model does not contradict the observed effect of intensive emission of "jets". It implies a possible emission by the compound system of a part of the matter as a light cluster with its subsequent decay into several particles. If the momentum of the light cluster is larger in comparison with its mass, then this group of particles looks like a jet. From the physical point of view the emission by the heated system of a part of its matter with its subsequent decay into components is natural. A well-known concrete example from the nonrelativistic nuclear physics is the process of fission of heavy excited nuclei with a subsequent emission of nucleons by fragments. It is known (see, for instance, ref. [6]) that the jets with given px are produced two orders as frequently as single particles, and their inclusive spectrum decreases slowlier than the particle spectrum. Such a behaviour (E(d3o/dp3))itt is caused by the following reasons: 1. The emission of a light cluster as a part of the matter of the system consisting of the quark-gluon plasma does not provide an immediate hadronisation. Consequently, in contrast with (5.4) for the particle production cross section, (E(d3o/dp3))iet should not contain a suppression factor caused by the nonequilibrium plasma. 2. The statistical weight of "jets" may be highly large, since the experimental conditions admit any combination of discrete quantum numbers of the jet, an arbitrary value of its angular momentum, etc. Now we evaluate very roughly the influence of these factors on the cross section of production of "jets". If one neglects the mass of the light (in comparison with its momentum) cluster, then {E(d3aldp3)^iet can be represented in the form analogous to (5.4): °°

0

Vm

-Vm

To estimate from below the statistical weight of the " j e t " we take into account its possible charge and angular momentum Gjet>STe jet -!7i iet .

(5-7)

426

B. N. KAIINKIN, The Process of Multi-Hadron Production

According to the charge there are at least three possibilities for the jet: e = 0, ¿ 1 , i.e. g ' > 3. Let the average value of the angular momentum of the "jet" be ( L ) . The corresponding statistical factors is g ' > 2 ( L ) + 1. To estimate the quantity ( L ) from below we consider a pair of 7r-mesons of the "jet" (addition of particles increases the number of possible combinations, and consequently gi iet ). Using the quasiclassical definition of the angular momentum, we get e

e t

L

h ( L ) a*

{ A p

n

) •

e t

2,

where (Ap„) ~ 1.3^ is an average value of the particle relative momentum under the assumption of isotropic decay producing a jet of the light cluster. 2(r„) ~ 1 fm is an average distance between 7t-mesons at the moment of their production. Assuming, that fj n ea 0.4 GeV/c (as in the majority of decays of cluster objects into 7r-mesons), we find that ( L ) sa 2. Therefore, G > 15. A direct calculation shows that the absence of the suppression factor also increases approximately by an order of magnitude and slows down decrease of the spectrum with increasing p . The estimates of the quantity ( E ( d a / d p ) ) from below, which have been obtained according to (5.6) and (5.7), and their comparison with experiment [54] are given in fig. 23. j e t

3

±

3

i e t

Fig. 23. Invariant cross sections of the production of "jets" (solid curve) and of a single meson (dashed line) 3

U

5

p^ / ( G e V / c )

4.

!

Correlation phenomena in the production of particles with large p^

The problem of associated multiplicity is important for any model of particle production with large pj_. It is shown in paper [11] on the basis of the principle of unitarity and analyticity that in the processes with large transfer momenta the associated multiplicities should grow maximally. The proof of this highly general statement allowed one to diminish the number of models which can pretend for describing the particle production with lage p±. The experimentally observed multiplicités in the events with a hadron with large turned out to be essentially larger than those averaged over all events at given energy E i a [55], this discrepancy increasing with growing E i n (see fig. 24). It was also shown [55] that the models of hard collisions encounter difficulties in describing this phenomenon. In the model [17] high multiplicities are naturally obtained in the processes with a large transfer momentum, since the dominating contribution to these processes in the hadron collisions comes from the total compound systems, i.e. the interactions with the coefficient of inelasticity close to unity (see fig. 2). • Before passing to the discussion of two-particle correlations with respect to momentum, let us recall the most useful quantities and their notation. A simplified scheme of

Fortschr. Phys. 32 (1984) 8

427

the processes in the e.m.s. of colliding hadrons is given in fig. 25. Particle " 1 " with large flying along the axis x, is registered at an angle of 90° to the collision axis. The projection of momentum of particle " 2 " onto the axis x is denoted by px. The dimensionless variable xe is defined as xe = px/px. A constituent of the momentum p2, which is perpendicular to the scattering plane of the particle " 1 " , is denoted by p0ut. An angle between the projection of the particle " 2 " onto the plane xz and the direction of the " 1 " particle motion is called the azimuthal angle : d h t d p

L

acg. t

d y d < p

~

( 2 n h c )

-

^

f

W

3

c dte-tl'HOiV)

o

V ( t )

J Vm

X

( W

-

m

c o r r

)

f ¿Vm

d y

0

c h

( y

-

y

0

)

e ^ i . « « » - ^ / *

J

Vm . | m"corr

J

d y

o h

0

( y

- y ' ) e ~ ^ I T

(5.8)

In (5.8) the rapidity y'' of the subsystem which received the recoil momentum p1M is defined by V' = 4 l n " f c ° r r ^

+

y",c°rr,

(5.9)

•C'corr — Pll.corr

where fll.corr =

m

e o r l

• sh

y

0

(5.10)

,

•Ecorr = i V o r r + P\M ch y0.

(5.11)

Es{ is the energy of the particle i with respect to this subsystem: E

s

(

=

-

p ^

0 I T

) y

0 0 ! !

(5.12)

.

Ei = m 1 ; ch y is the energy of the particle i in the lab. system; p is the projection of its momentum onto the direction of motion of the subsystem -

_

P.L

cos

(ft ~ 9")

P l t z

=

m

± i

Sh.y»tj.corr sh

yQ'

J

f v \

sh2 ycorr and

[j

c o t t

y

0

'

are the Lorentz factor and velocity of this motion:

ch y0' . Toon = m Lcorr ^corr /Scorr = yi-(l/ycorr) 2 .

(5.14) (5.15)

Let us consider some consequences of these correlations in the back hemisphere. Paper [55] has been devoted to the investigation of the distribution over rapidity y and azimuthal angle

3 GeV/c is formed. In fig. 26 the obtained in ref. [55] experimental data are compared with the results of calculation performed according to (5.8)—(5.15). The model reproduces correctly the observed tendences and also the absolute values of the" cross sections. It was mentioned [55] that the models of hard collisions fall to describe satisfactorily these characteristics too (averaged multiplicity of particles turns out to be too low, and the correlations too hard). The dependence of the particle yield in the back hemisphere on the azimuthal angle cp is shown in fig. 27, in which the results of the model are compared with the data of paper [56] obtained for two intervals of values of the particle transverse momentum.

429

Fortschr. Phys. 32 (1984) 8 /s = 63GeV

Q,rig = 90°

Ss=53GeV

150"? If)? 180°

0.6

Qtrig=90c

O-ty- 30° ,

150's^180°

0Â

°

S

0.2

k

0

S

0.4

o"

0.2

^

0

*

120°*!?!? 150°

90°*lfl- 120°

60°^90'

90%

120°

0Â 0.2 0 -h-2

0 2tU

-2 0 2*-U

-4 -2 0

2*-U-2

y

0 2 4

y

Fig. 26. Distributions over the rapidity of particles produced in events with the production of a hadron with ¡2 3 GeV/c in different intervals of the azimuthal angle q>

„

/F= 53

aj 13-pL -i—i^.

bj09iPl

OeV -1.9 OeV/c i 1.3GeV/c

c

•e

3

H . H\ 10°

20°

30°

Fig. 27. Dependence of the yield of mesons on Phhx

(6.12)

exceeding the value accessible by kinematics of hh-interaction. 6

) The critical remarks [69] towards this model are obviously due to the lack of information of the author of paper [69]. This is seen from the fact that the author refers to the first paper [70] only in which the model has been formulated in general and to the results of paper [65] which have been obtained by using incorrectly the model (it has been mentioned many times, see for instance ref. [23]).

Fortschr. Phys. 32 (1984) 8

437

Constraints (6.11) and (6.12) lead to important results: a) From (6.11) it follows that there is formed a complete compound system (CCS) containing the initially colliding hadrons. Unlike the case of pionization when a CS has the baryon number B = 0 for a CCS B = 2. If the constraint (6.12) is taken into account, B ^ 3. b) The cross section of formation of such a system a

c

•in

(6.13)

< a \ NN-

Comparison with experiment gives ac (0.20—0.25) c) The condition (6.12) means the choice of a channel with the realization in a CCS of a number of degrees of freedom small as compared to the case of pionization and emission of a particle at earlier stages of its development, i.e. at small times r ^ h/p f c

(6.23)

i.e., the path in the lab. coordinate system which the CS makes before hadronization should not be smaller than the minimal distance between nucleons in a nucleus (rc is the core radius in NN-interaction). At rc ~ 0.6 fm, rq ~ 0.5 fm/c for the initial proton energy Epin ^ 3 GeV. To conclude this section, we note that in the region of very small values of the invariant cross section of the cumulative-particle production an essential contribution to their yield may come from the fluctuon production mechanism. However, its realization requires further assumptions (and parameters) on the detailed structure of the nucleus and magnitude for local fluctuations of the density of nuclear matter. 3.

Multiple production initiated by the collision of relativistic nuclei

In the last years still more attention is paid to the multiple production in reactions between relativistic nuclei. The reasons are as follows: first, multiple processes are of increasing interest in general, second, in some research centres beams of accelerated nuclei are realized, third, there are hopes to obtain a new information on the process when the collective inelastic interaction of a large amount of nucleons becomes possible. To study this process for searching a new quality, it is desirable to have accelerated nuclei of a mass as large as possible and with an energy (energy per nucleón) at which even in the pp-interaction the multiple production is a completely developed process. Apparently, this requirement is fulfilled by energies of an order of 10—15 GeV/nucleon

Fortschr. Phys. 32 (1984) 8

441

and higher. Unfortunately, at Dubna and Berkeley accelerators a considerably lower energy is achieved. This makes it possible to investigate a number of interesting and important nuclear aspects of the reaction, however, for the study of the multiple processes the reached energies are not sufficient. This opinion was said already repeatedly (see, e.g., [75]). I t is also clear that the most evident manifestation of collective effects in the interaction of m a n y nucleon should be expected in "central" collisions of nuclei. Here we appeal to the cosmic data which though with a small accuracy because of a small statistics allow to judge the results of interaction of quite massive nuclei of the iron with nuclei of the photoemulsion, the bromine and silver at sufficiently high energies. The use of criteria developed in this methodics (see [70]) allows the separation and study of the cases of central collisions. Considering such a complicated process, we t r y to use again the picture of multiple production presented in the previous chapters. Generalizing the scheme of hadron-nuclear interactions we may expect t h a t the most probable channel of the reaction will be also the stripping and fusion into a unique CS of gluon fields of nucleons of overlapped parts of the interacting nuclei [77]. I n accordance with the above developed picture a part of the CS energy is realized as a longitudinal collective motion of subsystems of the CS matter. The remaining part of the CS energy is thermal. Valence quarks of nucleons realize leading baryon clusters, products of the "stripping" of an incident and aitarget nucleus. Their decay may be considered in analogy with [78]. The interaction kinematics is determined b y the energy-momentum conservation laws in a three-cluster systems. In the c.m.system we have |mpC2

+

+

1 . TbJ

{Niyi

+

N2y2) + Ecs =

Ecms,

T^ {N, j/ 7 l 2 - 1 + N2 ]/y22 - l) = 0,

Ecs = (k) [Ecms - m^N,

(6.24)

+ iV2)],

where TB ¡=«0.1 GeV and yu y2 are the temperature and Lorentzfactors of baryon clusters, Ecms, Nlt N2 are the energy in the c.m.s. of overlapped parts of nuclei and numbers of nucleons in them, (k) ~ 0.5 is the mean coefficient of inelasticity, m p is the proton mass, E c s the total CS energy. Generalizing the approach to the A A -interaction we meet at least two questions to be solved. The first concerns the definition of characteristics of the distribution over the rapidity of the CS matter. Following Chapt. I I we will define the thermal energy again by the relation TFQS = 6TECS, where dT = ym/sh ym (see (2.12)). However, it is to be decided whether one may use the previous parametrization of ym for the collision of nuclei. We think t h a t there is a quite serious reason to proceed in this way. Indeed, at the same specific energy the difference of quark rapidities in colliding nucleons is the same both in pp- and ^¿.-interactions, t h a t means the same maximum value of the forces of dynamical confinement of the colour. Consequently, the distribution of subsystems over rapidities in CS formed in ^.-interaction in the first rough approximation should be analogous to the one in pp-interaction (at the same specific energy). The second question arises while defining characteristics of the mean decaying configuration of CS formed in ^^.-interaction. I t is evident t h a t in this case the time of hadronization on the average is not a key parameter like in the elementary act. This can

442

B . N . KALINKIN, T h e P r o c e s s of M u l t i - H a d r o n P r o d u c t i o n

easily be understood if one takes into account that nuclei themselves are extended objects the volume of which is several times as large as the own total volume of nucleons. In nuclear collisions clusters of the quark-gluon matter localized first near the appropriate nucleons, initiators distributed over the nucleus volume are expanded over all directions and also towards each other overlapping and interacting between themselves. Thus, unlike the elementary act, for instance of pp-interaction, the quark-gluon plasma expands not into the vacuum but meets the matter practically in the same state (except for a zone near the nucleus surface). This should be expressed in the delay of hadronization time j h { A ) as compared to its value Th in pp-interaction. We do not know the magnitude of T h {A) . However, trying to establish a correspondence with the elementary act and to avoid an additional parameter we can make a reasonable assumption: at equal specific energies of the collision the CS hadronization occurs at the same temperatures and, hence, at the same energy densities as in the elementary act. Thus, the most important characteristics of the CS decay can be calculated with the use of relations of Chapt. I I .

30

q>

12

\

V

20

If*

10 i tii 10

i

tiM 100 (GeM/Af)

i

i i 1000

F i g . 3 8 . D e p e n d e n c e o n t h e incident-nucleus e n e r g y a) of half angles of t h e emission ,1/2, b) of t h e multiplicity density v = N±jninl

meson

Calculated results are in good agreement with the recent data on interaction of nuclei Fe - f Ag, B r at high energies. In Fig. 38 a we compare the calculated results and experimental data of half angles of emission of mesons versus the incident nucleus energy. In fig. 38b the comparison is given for data on the specific multiplicity v = Nn±/nint where

(6.25)

N„± = Ns — (z — 2 Na) Wint = 2.15(3 — 2Na

—

(6.26) Wnonint)

and Ns is the number of charged relativistic particles, z — the charge of an incident nucleus, Na the number of relativistic a-particles, nnoa ¡ n t — the number of noninteracting relativistic protons, emitted at angles 6 < 0.3 • p (p is the momentum of the incident nucleus'in (GeV/c)/nucleon).

443

Fortschr. Phys. 32 (1984) 8

Note also that according the relations (6.24) quantities 0Kil/2andr depend on the ratio of interacting (overlapped) masses of nuclei f = NJN 2 . In central collisions Fe + Ag, Br, £ ^ 0.75—0.79 that is close to that in pp-interaction. With decreasing f0Jr,1/2 and v are increasing. Qualitatively, this dependence explains the experimentally observed difference of the quantities d„il2 a n ( i v in central Fe + Ag, Br-collisions from the corresponding ones in collisions of protons and light nuclei with nuclei Ag Br, on the one hand, and the proximity of the given results to the case of pp-collisions, on the other hand (this is not shown in Fig. 38).

0.5

*Hk

OA

Cr»

V

0.3 0.2 0.1

0 Intg

Fig. 39

2 4 6/2

_1_J 10

6

L.

_l_lj L. 100

. 'I 1000

(OeV/A,1

Fig. 40

Fig. 39. Distribution of relativistic particles over the pseudo-rapidity r? = —In tan 8/z for different kinetic energies of nuclei Fig. 40. Amplitude of the distribution over the pseudorapidity An = ( N s j A r i ) a t a half angle of mesons as a function of the energy of an incident nucleus.

In Fig. 39 we compare the theory with experimental distributions of relativistic particles over pseudorapidity rj = —In tan 0/2 at different energies of incident nuclei, whereas in Fig. 40 the amplitudes A ^ 1/2 of these distributions at a half angle of emission of mesons (averaging is made over the interval Arj = ±0.6 relative to 0Jlilj2). So, the assumption of the collective nature of thé process of multiple production in nucleus-nucleus interaction and the use of the thermodynamical approach formulated in Chapt. II and III for the description of the decay of intermediate compound systems lead to results7) that do not contradict the observations, demonstrating thus uniqueness of the mechanism realized irrespective of the degree of complexity of initial conditions. 7)

At low initial energies (4—10 GeV/nucleon) the scheme expounded in this section is equivalent to the model of "nuclear pionization" (see [78] and references therein), i.e. its results are automatically reproduced.

4, Fortschr. Phys., Bd. 32, H. 8

444

B. N. KALINKIN, The Process of Multi-Hadron Production

VII.

Conclusion

1. From the whole consideration it is quite, clear that the last argument in the introduction against the extremism for the QCD is justified. I t has been shown that the statistic-thermodynamic scheme may well describe an extremely wide spectrum of characteristics in the " s o f t " and "hard" process regime in the whole energy range attained at present. Apparently, among modern models there is no analog which can achieve such an effectiveness in a large spectrum of solvable problems on multiple production: from an "elementary a c t " up to early stages in the hadron collective interaction (cumulative effect). I t is unlikely that this result though being preliminary is a pure accident; it rather testifies to the plausibility of the way projected. A starting point is the hypothesis of the dynamical mechanism of colour confinement switching on, as early as at short distances, high-power forces between colour charges scattering with ultra-relativistic velocities. Arising almost instantaneously, these forces excite many degrees of freedom of the field and form a statistic system at an early stage of the process. We gave grounds for this possibility using investigations of the behaviour of Yang-Mills classical systems. One of merits of the scheme presented is apparently a small number of free parameters used. The most important parameters are the mean time of hadronization rft and constant Cj in the formula for ym. The other model quantities may be defined with the use of its logic. For instance, values of r g and m corr were, in fact, determined by a simple argumentation based on known properties of multiple production. The' value of constant c 2 , by definition of ym, should equal the maximal mass of CS when there is no collective motion in it. Therefore, the CS should not contain more than one subsystem with mass mcorr (at ]/s a few GeV m corr & 2 2.5 GeV «a c 2 ). 2. The question may arise whether this demonstration of the statistical-thermodynamical-approach power does not mean another extreme, "thermodynamical extremism". We think that such a conclusion would be wrong. While developing the thermodynamical picture of the process we have in mind throughout the most probable channel of its evolution with a large energy release, i.e. the situation most specific for multiple production. Obviously, there should exist also cases when the process evolution proceeds beyond the presented scheme. Consider them in brief. a) Fluctuaticnally, with a small probability this may happen when in a system a minimal but sufficient for hadronization number of degrees of freedom is excited. For instance, nearby reacting quarks there may be produced a quark-antiquark pair with components which, first, will immediately complement the initial quarks to quasihadron colourless combinations and, second, will possess momenta (large enough) necessary for a real formation of a practically cold hadron. Then it may be expected that the process does not go by the thermodynamical scheme. I t is quite possible that just in this case the QCD methods are useful. In what region of kinematical variables one should expect a decisive contribution from this mechanism? A criterion though rather strict but indicating this region may be given in the case of production of particles with large p ± . Obviously, one should study the process requiring that x± — —> 1 i.e., near the kinematical limit. This criterion rejects contributions from mechanisms assuming the particle production as a result of the dissipation extended in time. Note that the most available experimental data do not allow us to judge of the behaviour of the inclusive cross section near the kinematical limit.

Fortschr. Phys. 82 (1984) 8

445

b) It is also clear that the statistic-thermo dynamicmechanism- should not work in the case of almost cold hadron systems, i.e., in the region of small energy release. First, at small excitations of the system the difference of quark rapidities is small, and consequently, dynamical forces of confinement are small. Second, such a system is essentially quantum. For it, by terminology of refs. [12, 13], the instability is not followed by mixing of trajectories, there exist discrete nonoverlapped levels. The very instability in a weakly excited quantum Yang-Mills system should not appear to the extent specific for the classical limit. Assuming quarks to be fermion we may apply a rough analogy with nonrelativistic nuclear reactions: In nuclei being Fermi-systems the region of small transfer momenta is strongly suppressed as a result of the Pauli principles. Consequently, this is one of possible reasons for weakening the effective interaction between quarks at small excitation of hadrons. Hence if follows that methods of the QCD type can be effective also for solving problems of the hadron spectroscopy (see, e.g., [79]) and especially correct in the case when the condition of adiabatic motion of quarks are fulfilled. A known example of the system of heavy slow quarks is the charmonium [80]. Thus, the general picture of inelastic processes cannot be drawn in a unique manner. This is due to a variety of properties of the hadron matter which appears in a different form under different conditions. The estimate we give here for the situation is, of course, rather approximate. It is necessary to make an extensive study of the problem which can determine in more detail macroscopic properties of the quark-gluon plasma and the nature of its phase transition into hadrons. / 3. To solve these problems is important not only for terrestrial purposes but also this may help in solving a number of jfroblems in cosmology (early stages of evolution of the Universe) and astrophysics. In the last years these problems are given a good deal of consideration. We may point to the interesting work by S. WEINBERG "After the first three minutes" where the authors considers a possible change of' 'eras" curing the first three minutes of the existence and evolution of the Universe. In particular, considering the hadron era the author, tending to show the possibility of a phase transition of the quark-gluon matter into hadrons, applies to ingenious arguments based on QCD and an analogy with the theory of magnetic. However, he says that it may happen to be difficult to prove rigorously the existence of the above transition and, moreover, that we cannot calculate the temperature reliably. Roughly speaking, it should be of an order of A, but it will not be surprising, if it turns out to be of an order of 2A or of A/2. Perhaps, the most definite conclusion in [81] concerning the hadron era is that the phase transition takes place in a finite range of temperatures of the order AT m 120 MeV. These difficulties are not surprising as in macroscopic processes when there is a massed transition of the quark-gluon plasma into a large number of soft hadrons (mainly, 7t-mesons), the QCD methods are insufficient. On the contrary, the results given here testify the possibility to judge, even now, more definitely about the existence of a phase transition characterizing the start of hadron era and its limits. In Chapt. IV and V we have mentioned that in calculating the invariant inclusive cross section of production of particles with large Pj_ the temperature of CS emitting a hadron at early stages at t < rq has been defined from the relation energy density for the quark-gluon plasma, while at t > rq for the hadron gas. We illustrate this in more detail in fig. 41 [28], Let the change of heat capacity due to the phase transition occur in some moment r ( . The curve with index r t = 0 represents the calculation under the assumption of in4*

446

B. N. KALINKIN, The Process of Multi-Hadron Production

stantaneous formation of the hadron phase in a CS just created. It is seen that it well describes the experiment near small p x and is in disagreement with it at large p ± . The case T( —> OO means the assumption that hadrons are emitted by a CS that is always in the state of quark-gluon plasma. Here we observe an opposite situation: being in agreement with experiment at large Pj_ (early stages) Ed3a/dp3 is in large disagreement with it at small p±. From fig. 41 it is seen that'the best agreement is achieved at xt = rq a 0.6 fm/c. For comparison we show also results corresponding to intermediate cases: xt — 0.3 fm/c and r ( = 1.2 fm/c.

J

I

L

1

3

5 OeV/c)

/(

Fig. 41. Invariant cross section of the production of n + mesons in pp-collisions at Epin = 200 GeV

The results indicate rather clearly the manifestation of two phases of the hadron matter. It may be considered that up to about t m rq the matter is mainly in the quarkgluon phase, and at T9 5S t SI xh in the quasihadron state the transition of which into real hadrons is practically completed by a moment t = rh. Consequently, the transition of the quark-gluon phase into the hadron one occurs in the temperature interval T(t„) -r- T{th). To be specific, at CS masses of the order ]/s = 30 GeV these temperatures approximately equal: T(rq)

~ 0.40 GeV

and

T(rh)

=

0.16 GeV.

Correspondingly the CS energy densities are: e(r„) ~ 8.3 GeV/fm 3 and

e{rh) ~ 0.54 GeV/cm 3 .

Thus, the average critical temperature of the transition and its smearing can be estimated by Te ~ (0.23 ± 0.07) GeV. As a result, parameters specifying the start of the hadron era in the evolution of the Universe turn out to be directly connected with recent terrestrial experiments on multiple production. These results should, of course, be further verified and enlarged.

Portschr. Phys. 32 (1984) 8

447

References [1] BLATT, J . M., WEISSKOPF, V. P., Theoretical Nuclear Physics, New York, 1952. [2] DAVYDOV, A. S., Theory of Atomic Nucleus, Fizmatgiz, Moscow (1958). [3] Photonuclear Reactions, Moscow, IL (1953). [4] EYGES, L . , P h y s . R e v . 8 6 (1952) 3 2 5 .

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BLOKHINTSEV, D. I., Zh. eksper. teor. Piz. 32 (1957) 350. TYAPKIN, A. A., Elementary Particles and Atomic Nuclei 8, 3 (1977) 544. FEYNMAN, R. P., et. al., Phys. Rev. D 18, 9 (1978) 3320. RANFT, G., RANFT, J., Elementary Particles and Atomic Nuclei 10, 1 (1979) 90.

[9] EFREMOV, A . V . , RADYUSHKIN, A . V . , R e v i s t a N u o v o C i m e n t o 3 ( 1 9 8 0 ) . [ 1 0 ] AZIMOV, YA. I . , e t a l . , U s p e k h i . P i z . N a u k 1 8 2 , 3 (1980) 4 4 3 ; DOKSHIZER, Y U . L . , e t a l . ,

Lecture in LIYaP XIII Winter School, LIYaF, Leningrad (1978); DOKSHIZER, YU. L., et al., Lecture in XIV LIYaP Winter School, LIYaP, Leningrad (1979). [ 1 1 ] LOGUNOV, A . A . , MESTVIRISHVILI, M. A . , PETROV, V . A . , Y a d e r n . P i z . 3 1 (1980) 4 8 7 . [12] CHIRIKOV, B . V . , P r e p r i n t I Y a S . O . A N S S S R , 7 8 - 6 6 , N o v o s i b i r s k ( 1 9 7 8 ) ; CHIRIKOV, B . V . ,

SHEPELYANSKI, D. L., Pisma Zh. eksper. teor. Fiz. 34 (1981) 171. [13] MATINYAN, S. G., et al., Zh. eksper. teor. Piz. 80 (1981) 830. [14] CHETYRKIN, K. G., et al., Preprint IYalAN SSSR P - 0126, Moscow (1979). [15] BOGOLUBOV, N. N., SHIRKOV, D. V., Introduction in Theory of Quantized Fields, Nauka, M o s c o w ( 1 9 7 8 ) ; VLADIMIROV, A . A . , SHIRKOV, D . V . , U s p e k h i . F i z . N a u k 1 2 9 (1979) 4 9 7 .

[16] COMBRIDGE, B. L., KRIPFGANZ, J., RANFT, J., Phys. Rev. Lett. B 70 (1977) 234. [17] KALINKIN, B . N . , J I N R P - 2 - 8 1 - 7 2 9 , D u b n a ( 1 9 8 1 ) ; SHMONIN, V . L „ H F P I 71-14, A l m a - A t a (1981).

[18] KOGUT, J., SUSSKIND, L., Phys. Rev. D 12 (1975) 2821. [19] BORN, M., OPPENHEIMER, J. R., Ann. Phys. (Leipzig) 84 (1927) 457. [ 2 0 ] KOGUT, J . , SUSSKIND, L . , P h y s . R e v . D 9 (1974) 3 5 0 1 . [ 2 1 ] WILSON, K . , P h y s . R e v . D 1 0 (1974) 174.

[22] BAROS, A. J., Rev. Mod. Phys. 52 (1980) 199. [23] KALINKIN, B. N., SHMONIN, V. L., Elementary Particles and Atomic Nuclei 11, N 3 (1980) 630. [ 2 4 ] KALINKIN, B . N . , CHERBU, A . V . , SHMONIN, V . L . ,

JINR

P2-11620,

Dubna

(1978);

Acta

Phys. Austr. 60 (1979) 165. [25] KALINKIN, B. N., CHERBU, A. V., SHMONIN, V. L., J I N R P2-12330, Dubna (1979); Phys. Scr. 21, (1980) 979. [26] KALINKIN, B. N., SHMONIN, V. L., J I N R P2-80-727, Dubna (1980). [27] KALINKIN, B. N., SHMONIN, V. L., J I N R P2-80-176, Dubna (1980). [28] KALINKIN, B. N., SHMONIN, V. L., J I N R P2-80-794, Dubna (1980). [ 2 9 ] KRZYWICKY, A . , WEINGARTEN, D . , P h y s . L e t t . 1 3 5 0 (1974) 2 6 5 ; CHAO, A . V . , QUIGG, C.,

P h y s . R e v . D 9 (1974) 2016. [30] HAMA, M . , NAGASAKI, M . , SUZUKI, H . , P r o g r . T h e o r . P h y s . 5 7 (1977) 160.

[31] ADAMOVICH, M. I., et al., P I A N SSSR, 108, 3 N a u k a , Moscow (1979).

[32] KALINKIN, B. N., SHMONIN, V. L., Phys. Scr. 24 (1981) 498. [33] ALBINI, E. A., et al., Nuovo Cimento A 32 (1976) 101. [34] TOME, W., et al., Nucl. Phys. B 129 (1977) 365. [35] BRANDELIK, R . , et al., P h y s . L e t t . B 89 (1980) 418.

[36] DERADO, I., et al., Nucl. Phys. B 143 (1978) 40. [37] NEZRICK, F . A . , FERMILAB, C o n f . 7 7 / 1 2 , E x p . B a t a v i a , 1977.

[38] ANTINUCCI, H., et al., L e t t . N u o v o Cim. 6 (1973) 121.

FERBEL, T., in Proc. I l i Internai. Colloquium on Multiparticle Reactions, Zakopane, 1972. [39] WOLF, G . , D E S Y 7 9 / 4 1 , H a m b u r g , 1979.

[40] BARBER, D. P., Phys. Rev. Lett. 43 (1979) 901. [41] AZIMOV, Ya. I., et al., XY LIYaF Winter School, LIYaF, Leningrad (1980). [42] KINOSHITA, K., K E K N 33 (1979) 144. [43] KALINKIN, B. N., SHMONIN, V. L., J I N R P2-80-145, Dubna (1980). [44] DE BRUIJN, N. G., Asymptotic Methods in Analysis, North-Holland Pubi. Co. — Amsterdam, P. Nordhoff LTD-Groningen (1958).

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B. N. KALINKIN, The Process of Multi-Hadron .Production

[45] MENG TA CHUNG, P h y s . R e v . D 9 N 11 (1974) 3 0 6 2 .

[46] KALINKIN, B. N., SHMONIN, V. L„ Phys. Scr. 24 (1981) 498. [47] DEELL, S., YAN, T. H., P h y s . R e v . L e t t . 25 (1970) 316.

[48] LEDERMAN, L. M., Proc. 19TH Internat. Conf. High Energy Phys., Tokyo, 1978. [ 4 9 ] ANTEEASYAN, D . , e t al., P h y s . R e v . D 1 9 ( 1 9 8 0 ) 7 6 4 .

[50] ABBAMOV, V. V., et al., I H E P , 80-80, SERBE-100, Serpukhov, 1980. [51] KOTJBKOTTMELIS, C., et al., P h y s . L e t t . 81 B (1979) 405.

[52] ALPEK, B., et al., Nucl. Phys. B 100 (1975) 237. [53] COTTBELL, R,, et al., Phys. Lett. B 55 (1975) 341. [54] BKOMBEEG, C., et al., P h y s . Rev. L e t t . 4 3 (1979) 565. [55] DEBADO, I., et al., Nucl. P h y s . B 143 (1978) 40. [56] DABBIULAT, P . , et al., Nucl. P h y s . B 107 (1976) 429.

[57] BUSSEB, P. W., et al., Nucl. Phys. B 107 (1976) 1. [58] SOSNOWSKI, R., Proc. 18 th Internat. Conf. High Energy Physics, J I N R , Dl, 2-10400, Dubna, 1977. [59] FISK, R. J., et al., Phys. Rev. Lett. 40, N 15 (1978) 984. [60] DELLA, M., HEGEA, et al., Nucl. Phys. B 127 (1977) 1. [61] CLABK, A. G., Nucl. Phys. B 160 (1979) 397. [62] NIKOLSKI, S. I., V Internat. Seminar High Energy Phys., Dubna, 1978. [63] KALINKIN, B . N . , SHMONIN, V . L . , Z . P h y s . C 5 (1980) 121, P r e p r i n t H E P I - 8 0 - 1 0 , A l m a - A t a , 1980.

[64] ANDEBSON, R. L., et al., Phys. Rev. Lett. 38 (1977) 263; CAMEBINI, V., et al., Phys. Rev. Lett. 85 (1975) 483. KNAPP, B., et al., Phys. Rev. Lett. 34 (1975) 1040. [65] GULAMOV, K. G., et al., Yad. Fiz. 26, 1095 (1977). [66] TOLSTOV, K. D., et al., J I N R P2-6897, Dubna (1973). [67] GAGAEIN, YU. F„ et al., Izv. AN SSSR, ser. fiz., 38 (1964) 988. [68] KALINKIN, B. N., SHMONIN, V. L., J I N R P2-7871, Dubna (1974). [69] NIKOLAEV, N. N., Uspekhi. Fiz. Nauk 134, 3 (1981) 370. [70] KALINKIN, B . N . , SHMONIN, V . L . , Y a d . F i z . 2 1 (1975) 6 2 8 .

[71] BALDIN, A. M„ Santa Fe Conf. High Energy Physics Nucl. Structure, 1975. [72] KALINKIN, B. N., CHEBBTT, A. V., SHMONIN, V. L., Fortschr. Phys. 28 (1980) 35. [73] BALDIN, A. M., et al., J I N R 1-12396, Dubna, 1979. [74] HIKIFOBOV, N . A . , e t al., P h y s . R e v . C 2 2 (1980) 7 0 0 .

[75] WILLIS, W., CERN Courier 22, January/February, 1982. [76] VABYUKHIN, V . V . , e t a l . , F T I A N S S S R , N 6 1 6 , L e n i n g r a d (1979).

[77] VAEYUKHIN, V. V., et al., Pizma Zh. eksper. teor. Fiz. 35, N 6 (1982) 261. [78] KALINKIN, B . N . , e t a l . , P h y s i c a S c r i p t a 2 1 (1980) 7 9 2 . [79] GEEASIMOV, S. B . , GOVOBKOV, A . B . , J I N R P 2 - 8 1 - 5 3 8 , D u b n a , 1981.

[80] VAINSTEIN, A. I., et al., Uspechi Fiz. Nauk 123 (1977) 217. [81] WEINBEBG, S., Phys. Scr. 21 (1980) 773; WEINBEEG, S., Uspechi Fiz. Nauk 134, No. 2 (1981) 333.

Fortschr. Phys. 32 (1984) 8, 4 4 9 - 4 7 2

Poincaré Gauge Theory, Gravitation, and Transformation of Matter Fields Wolfgang

Drechsler1)

Max-Planck-Institut für Physik und Astrophysik, Werner-Heisenberg-Institut für Physik, München, Federal Republic of Germany

Motto: The question whether Kaluza's formalism has any future in physics is thus leading to the -more general unsolved main problem of accomplishing a synthesis between the general theory of relativity and quantum mechanics. W. Pauli Theory of Relativity, 1958

Abstract In the generalized Kaluza-Klein theories a (4 + 2tf)-dimensional Riemannian space is used as a configuration space to unify gravitation with a noriabelien gauge symmetry characterized by a gauge group 0 of order N. The original metric space Vi+N, however, is broken down in these theories to a local product structure Mi X VN (Mi being Minkowski space-time) leading thereby to a description in terms of a fiber bundle over space-time Vi with structural group G. In this paper we base the discussion from the very beginning on a bundle structure as the underlying geometric stratum and treat the quantum mechanical aspect of matter by representing it in the form of a generalized wave function, X), defined as a section on a fiber bundle associated to a principal G-bundle over space time having a homogeneous space of the group G as fiber. To make contact with gravitation we consider various soldered bundles possessing the Poincaré group, G = 180(3, 1), as structural or gauge group and define Poincaré gauge fields as generalized matter fields on them. After reviewing the formulation of general relativity as a gauge theory of the Lorentz group we turn to the affine case and present a geometric formulation of a Poincaré gauge theory based on a Riemann-Cartan space-time U i with axial vector torsion. Two sets of field equations are discussed relating the material source quantities to the geometric fields. Besides Einstein's equations coupling the metric to the classical distribution of energy and momentum of matter a second set of nonlinear field equations is set up relating a bilinear current in the Poincaré gauge fields to the axial vector torsion field of the underlying space-time geometry. Finally, a breaking of the Poincaré symmetry is introduced in a manner involving the generalized matter field (x, X) freezing thereby the translational gauge degrees of freedom, however, leaving the Lorentz gauge symmetry unaffected.

I.

Introduction

A variety of proposals have recently been discussed in particle physics extending the basic geometric framework within which a dynamics is formulated to higher dimensional spaces. One considers theories formulated in spaces of 4 + N dimensions. However, the way the additional N dimensions are treated geometrically is quite different in the various theoretical schemes. The common aim o.f all these endeavours is the hope that with the help of the additional dimensions and the associated degrees of freedom a method could be established "for a differential geometric description of the internal Talk presented at the X I I . International Conference on Differential Geometric Methods in Theoretical Physics, Clausthal 1983.

450

W. DRECHSLER, Poincaré Gauge Theory

dynamics of extended hadronic particles — with hadrons being the constituent building blocks of matter as we observe it in nature. The idea thus is that the additional dimensions come into play at small subnuclear distances determining the dynamics in an essential way without requiring the explicit introduction of subunits of hadrons. One group of theories with more than four dimensions uses a generalization of the idea first introduced by KALUZA (1921) and KLEIN (1926, 1958) [compare also EINSTEIN and BERGMANN (1938)] who successfully tried to unify gravitation and electromagnetism in one geometric framework. The proposal was to give a combined geometric description of gravitation and electromagnetism in a five-dimensional metric space with the components gr,,5; (i = 0, 1, 2, 3, of the metric playing the role of the electromagnetic potentials, A ^ i.e. /&» + g^2A„Av

gyJAA

\

055 /

ffnPA*

where is a constant with the dimension of a length (KLEIN (1958)). Considering not gravitation and electromagnetism but gravitation together with an interaction characterized by a nonabelian gauge group of order N one can follow the same procedure [KEENER (1968), CHO ( 1 9 7 5 ) ; see also ORZALESI (1981) and WITTEN (1981)] and unify a nonabelian gauge interaction with gravity by starting from a metric description in a configuration space having 4 + N dimensions, with the metric tensor 9UB breaking down into blocks according to (fi, v = 0, 1, 2, 3; a, b = 1, 2, 3 . . . N; AM° denotes the gauge potentials, and gab is the Cartan-Killing metric of the gauge group which is supposed to be semi-simple): gab = In principle, every field occuringin the formalism depends on all the 4 + N coordinates xA = {x", x°);

fi = 0 , 1 , 2 , 3 ;

A = 1,

2...N.

Posing, however, additional constraints (generalized "cylinder conditions") would eliminate the dependence of some of the fields on the additional N coordinates which we refer to as the internal coordinates in the following. In the original Kaluza-Klein theory there was only one additional coordinate, called x 5 , which was supposed to vary in a compact one-dimensional space leading thereby to a periodic dependence of matter functions on x5. I t was, moreover, assumed that the g^ and A^ did not depend on the fifth coordinate and that g5i was a constant [essentially taken to be equal to —1 with the signature of the V5 being (-| )]2). This was called the cylinder condition. Thereby the five-dimensional Riemannian space was, in fact, broken down to a ?7(1) fiber bundle over a four-dimensional space-time with metric g^ and bundle connection with coefficients A^. I t was also proposed [JORDAN (1947), THIRY (1948), compare also LICHNEROWICZ (1955)] to set the 55-component of the fivedimensional metric equal to a scalar field dependent on xp, ju = 0 , 1 , 2, 3; which is then seen t o p l a y t h e role of a B r a n s - D i c k e field. [BRANS and DICKE (1961)].

One can now proceed similarly in a (4 + .^-dimensional theory as was done before in the original Kaluza-Klein theory. The breaking down of the Riemannian metric in 4 + N dimensions to the ordinary metric gflv of space-time and the metric gab of an internal space as well as the off-diagonal metric components describing the gauge aspect 2

) Compare also EINSTEIN (1927). For the projective interpretation of the five-dimensional theory see PAULI (1933) and the literature cited there.

Fortschr. Phys. 32 (1984) 8 of the the theory has been called spontaneous compactification

451 [SCHERK and SCHWARZ

( 1 9 7 5 ) , CREMMER a n d SCHERK ( 1 9 7 6 ) , SALAM a n d STRATHDEE ( 1 9 8 2 ) ; c o m p a r e also CHO

and FREITND (1975)]: The space Vi+N is assumed to suffer a local product decomposition into M4 X VN, where is Minkowski space-time (isomorphic to the local tangent space, Tx, of a pseudo-Riemannian space-time F 4 ), and VN is a A7-dimensional internal space (usually regarded to be a compact space). The physical idea is that the distances characterizing the additional N dimensions in V4+N are small when viewed from ordinary spacetime, and the aim is to relate these dimensions to the domain of an internal dynamics of particles at subnuclear distances. The detailed dependence of the various geometrical or matter fields on the additional internal coordinates is, however, quite unclear in these generalized Kaluza-Klein theories. Usually one introduces ad hoc conditions (generalized "cylinder conditions") suppressing arbitrarily the dependence on the additional variables for various quantities [CHO and FREUND (1975)]. In other models one only discusses the so-called zero frequency modes for the internal motion and disregards higher order modes in an harmonic analysis on VN. The whole procedure results in an effective bundle description tif the underlying geometry after spontaneously breaking the original (4 N)dimensional metric geometry introducing "cylinder conditions" a la Kazula-Klein. Even if the higher excitations in an internal mode expansion were kept, one would still have a situation where the Riemannian space Vi+N is broken down to a local product structure, i.e. to a certain fiber bundle over the conventional 4-dimensional space-time of general relativity. This reduction of symmetry of the underlying (4 -+- 2V)-dimensional configuration space could more appropriately be called a spontaneous fibration. A further question in this context is whether it is possible to characterize in an unambiguous manner the internal space (the fiber) by a length parameter and how big this length actually is; whether it is of the order of the Planck-length, R = _RP1 zv 10" 33 cm [compare K L E I N ( 1 9 5 8 ) ] , or w h e t h e r i t is of c o n s i d e r a b l e d i f f e r e n t order, R & ( 1 0 ~ 1 3 — 1 0 ~ 1 6 ) c m

being associated with strong interactions.

That a fundamental theory of matter should contain an elementary length parameter is

a n old i d e a [ c o m p a r e HEISENBERG ( 1 9 3 8 , 1 9 6 7 ) , LANDAU (1955)]. W h e t h e r s u c h a l e n g t h

would indeed, appear in the formalism as a result of a spontaneous fibration in a KaluzaKlein-type theory is quite another question. Probably it is wiser to start right away from a particular fiber space raised over a (pseudo)-Riemannian space-time, F 4 , or (in the presence of torsion) over a Riemann-Cartan space-time, Uit or even over a RiemannCartan-Weyl space, W4, avoiding thereby the use of a metric description in a higher dimensional space with the immediate need to break this Riemannian space down to a bundle over a F 4 with a fiber, characterized by a length parameter, providing the arena for the internal dynamics. Models based directly on fiber spaces over space-time (with the base possessing possibly additional geometric structures beyond that derivable from a Riemannian metric) have been considered before [DRECHSLER (1975, 1976, 1977, 1979, 1 9 8 2 ) , c o m p a r e also DRECHSLER a n d MAYER ( 1 9 7 7 ) ; SMRZ ( 1 9 7 9 ) , a n d STELLE a n d W E S T

(1980)]. The fiber in these bundle geometries is taken to be the homogeneous space of some group G, i.e. F = QjH with H being a subgroup of G and with the fiber F being characterized by a fundamental length parameter R of the order of 10~13 cm [DRECHSLER (1975,1976)]. Particularly interesting is the situation where G = SO{4,1) and H = SO{3,1) with the coset space GIH being a (4, l)-de Sitter space V4' of radius R providing the geometric arena for a gauge dynamics based on the (4, l)-de Sitter group. In this context one can, moreover, identify the stability subgroup H operating in the fiber for each spacetime point with the local Lorentz group operating on the Lorentz frame boundle over space-time since one considers a soldered3) bundle over space-time possessing the (4,1)de Sitter group as structural group; i.e. yielding, due to the soldering, a geometric 3

) Compare EHRESMANN (1950), KOBAYASHI (1957) and LICHNEROWICZ (1962).

452

W. DRECHSLER, Poincaré Gauge Theory

framework with a noncompact semi-simple gauge group and an elementary length parameter B built into it. Contracting the de Sitter group to the Poincaré group in letting R go to infinity, one has a scheme in which the Poincaré group, ISO (3, 1), acts as a gauge group. We shall come back to a theoretical framework treating the Poincaré-group as a gauge group in detail below in connection with gravitation. Here, in closing the introduction, we would like to add some general remarks concerning the representation of matter in such a formalism. In all these theories, basing the description on the geometry of some fiber bundle over space-time — either by specifying a particular bundle directly or by obtaining a local product structure through a spontaneous breaking of a higher dimensional metric space as in the generalized Kaluza-Klein theories — there will, always appear "generalized wave functions" for the description of matter which in general depend on more than the usual four space-time coordinates if the higher dimensional theories are, indeed, taken seriously. These generalized wave functions represent sections FAB, F, G) on F„=G/H

a : B —- E (B, F, G ) Fig. 1. Generalized matter wave function as a cross section on a fiber bundle E(B, F, 0) over space-time with structural group 0

bundles E(B, F, G) associated to a principal Cr-bundle over space-time (compare Fig. 1). [DRECHSLER 1977 C], T h e y c a n b e w r i t t e n as

= 4>(x, X); x = (x^; [i = 0, 1, 2, 3) € B 1 = (1«;

a

(space-time)

= 1 ... N) e Fx = G\E

(fiber).

We here continue to denote the dimension of the internal space by N which now is a number smaller than (or equal to) the order of the group O since N = dim F = dim G — dim H. I t is interesting, as mentioned above, to consider bundles with homogeneous spaces associated with some subgroup H of G as fiber which, moreover, possess a first order contact with the base space; i.e. to take a soldered bundle. Such bundles are usually considered in the context of general relativity; however, one normally disregards soldering in connection with a gaugte formulation of internal symmetries as they are discussed in particle physics. There the internal space is an abstract space describing chargetype degrees of freedom which are unrelated to space and time. Here we thus consider soldered bundles over space-time with a transitive action of the gauge group, i.e. G acts as a group of motion in F. If the gauge symmetry of a theory based on the soldered bundle E(B, F = GjH) is spontaneously broken to the Lorentz subgroup H of G one can suppress the dependence of the generalized matter wave function on the internal coordinates by using a non-

Fortschr. P h y s . 3 2 (1984) 8

453

linear realizations of the gauge symmetry (? representing only the subgroup H in a linear manner [TSEYTLIN ( 1 9 8 2 ) ] . We shall discuss in the following a gauge theory of the Poincaré group in connection with gravity describing matter in terms of a wave function

(z, X) and their adjoints and obtain a Lagrangian density lf{x) of the usual type by integrating ,¥{x, X) in a gauge invariant manner over the local fiber, i.e. over the internal coordinates, averaging therby over the internal degrees of freedom [DRECHSLER ( 1 9 7 7 B ) ] . This might be interesting in order to study the theory in a projection valid for point-like objects undergoing a translational motion in the base space (space-time) of the original bundle which, in a'geometric manner, describes the gauge dynamics of extended objects characterized by a fundamental length parameter R. We shall, however, not use a Lagrangian formalism in our subsequent discussion for deriving the field equations but rather prefer to motivate these equations by basing them directly on geometric reasonings. This, of course requires an additional check on the consistency of the equations which would be automatically guaranteed in a Lagrangian derivation. After these general remarks we now investigate a specific geometric framework which allows the Poincaré group to play the role of a gauge group and investigate what behaviour the matter fields would have in such a scheme and how they would have to transform under the gauge symmetry group.

II.

Classical General Relativity as a Gauge Theory of the Lorentz Group

We first consider classical general relativity which is based on a pseudo-Riemannian space F 4 with metric g^ and metric connection

W . D r e c h s l e r , Poincaré Gauge Theory

454

and the metric tensor obeying V/j^ = 0 (Greek indices refer to a natural (holonomic) basis, d^fi = 0 , 1, 2, 3, in Tx, the local tangent space at x € F 4 ; similarly, dx^ ; ¡t, = 0, 1, 2, 3, is a corresponding natural basis of one-forms in the cotangent space Tx*. Quantities derived from a Riemannian metric and covariant derivatives with respect to the metric connection are denoted by a bar; below we shall also consider nonmetric connections). Matter enters Einstein's equations through the symmetric energy-momentum tensor T w h i l e the geometry is described through the Einstein tensor, constructed from the contractions of the Riemannian curvature tensor R^ex of the space-time F 4 : Guv = Hp, — — g^R — xT

.

(1)

Here the barred (i.e. metric) quantities are R^ = P [.lov e — Vpi r) T °

a 4- T ,»f ftQ* T fif I X [LA

X

i

= RliQvage", with

1 QV f x,«fOA - * pv

being the Riemann-Christoffel tensor obeying the symmetry relations RfiQva

=

Rgfiva —

Rpeov

=

RfafiQi and

R =

R^g^".

As is well-known, the left-hand side of Eqs. (1) follows from the variation of the metric in the Einstein-Hilbert action L = (1/2*) f R^—g d*x with g = det (g^) and x denoting Einstein's gravitational constant (x = — SnK/c*; K = Newton's gravitational constant). Moreover, there are Bianchi identities for the components of the curvature tensor to be obeyed (see below) guaranteeing that G^ is covariant divergence-free i.e. that =

0.

In Einstein's equations (1) the classical energy-momentum distribution of matter is acting as a source for the geometric fields. Usually a tensor T ^ is introduced describing the flow of a number of classical pointlike particles or the motion of a fluid. There are usually no fields introduced for the classical description of matter. Now, can this classical geometric theory of gravitation be interpreted as a gauge theory, and if so, what is the relevant gauge group? Is it the Poincaré group; is it the Lorentz group, or are even the translations alone the essential symmetry transformations in a gauge interpretation of the gravitational interaction? All three possibilities have, indeed, been suggested in the literature. Let us first observe that the Poincaré, Lorentz or translational gauge degrees of freedom (if they are there) seem to be hidden in Einstein's theory. All what is apparent is the general covariance of the equations against changes of an atlas {Ut, 4>i} on the space-time manifold F 4 with the map , : Ui Di cz ikf4 defining a local coordinate system in Vi ci F 4 for all the TJi ; i = 1 ... n, covering F 4 and inducing a so-called natural system of axes, 8¡t. Furthermore, there is the possibility of transforming the Christoffel symbols rn V e locally to zero (equivalence principle). O n t h e other hand, i t has been shown b y E l i e C a r t a n [ C a r t a n (1922, 1924, 1935)] 4 )

that even from a point of view of pure geometry — i.e. disregarding the representation of matter in the theory — it is interesting to refer all quantities determining the geometry of a Riemannian space to a local orthonormal (i.e., in a 4-dimensional space-time, Lorentzian) system of axes obtained by a transformation with the vierbein fields, 5 ) g^ = ¿¿¿Svit;

Vik = diag (1, - 1 , - 1 , - 1 ) ,

ds2 = g„, dx" ® dx" = 6° (X) 0° 4) 5)

01 ® 0 1 -

02 (X) 0 2 -

(2) 03 (X) 9s,

(3)

Compare in this context also W e y l (1929) and Sciama (1962). W e use the summation convention with regard to the greek (holonomic) and latin (anholonomic) indices ¡x, v = 0, 1, 2, 3; i, k ... — 0, 1, 2, 3, respectively.

Fortschr. Phys. 82 (1974) 8

455

where 0* = A / D X " ;

¿ =

0,1,2,3

are the fundamental one-forms of the space providing a base in the dual tangent space, T x * , a t the point x 6 F 4 . The inverse fields, A/, obeying A/A,,» = ik = —wki on the Lorentz frame bundle 2>(i74) over f74 which now has a metric part, wik and a torsion contribution, Tik; the latter having tensorial transformation character under Lorentz gauge transformations. Thus from now on we make the following replacements [compare DKECHSLER (1982a)] 11 ) V4 ->

Ut,

a>ik -s- u>ik = Wik + rjik

rjik = r

F->

k>

or >

in matrix notation [see Eq. (13)]

+ k^h, = —Quak)} +

m

1

cu = cu + r , (31)

^w»*))

V.

Here V denotes the covariant derivative with respect to the connection co which can be split into a metrix covariant derivative, V, and a torsion contribution. The compatibility of metric and torsion is expressed by the equations 17egfl, = 0 which, due to the antisymmetry of the tensor = Xjl^X^Ky^ in its last two indices, is equivalent to the equations F 7 ^,. = 0 valid in a F 4 . Cartan's structural equations for a Riemann-Cartan space-time now read [with 0 as defined before; see Eq. (13)] 76 = dd -

co

6 = r,

dm — co /\ co = Q.

(32) (33)

Here t = I : I is a column vector of torsion two-forms w3/ =

fP /x 6%*,

(34)

where S j f = — S i f is the torsion tensor [with (1/2) = K j i k = —K j k i being the torsion contribution to the connection coefficients, as introduced in (31) above]. Furthermore, Q = {Q l s ) is a matrix of curvature two-forms with Qit = ruPK

= j

0" /v 6 = m + r in Eq. (33). The result is Q = Q + Q' 11 )

For the third line compare Eq. (8) and Eq. (34) below.

5

Fortschr. Phys., Bd. 32, H. 8

(36)

460

W . DRECHSLER, Poincaré Gauge Theory

where Q is given by Eq. (15) and Q' has the form Q' = di — 53

i

—

T/\CU —

r ^ r

=

Vx

— T A T ,

(37)

For the components of the incurvature tensor this implies (with xik = —X^ = 6'Kjih) Rklis = Rklis + Pklis >

(38)

PkUs = VkKus — V iK k i s + KkijKiJ — K u j K k s i , with the latter tensor having the symmetries Pkits = —Pms — —PkisiThe Bianchi identities following from Eqs. (32) and (33) read v% = di -

co

6,

(39)

+ Q^co=0.

(40)

i = -Q

VQ=dQ-m^Q

Expressed in terms of the tensors Sjik and Rkiis they take the form

(40')

V{sRim=8{JRmi.

W e now consider an affine base in Tx, the local tangent space at a: of a RiemannCartan space-time, C74, and introduce the row matrix with five entries (compare Eq. (13)) : E = (x, 8).

(41)

E represents an orthonormal frame e,- ; j = 0, 1, 2, 3, the origin of which is displaced in T x from the point of contact a; € i74 to x = e^x1 = —e^x1. Here we have represented the Lorentz invariant quantity x 12 ) either in terms of components referring to a Lorentz base e; with origin at a; £ E74, or to a displaced and Lorentz rotated frame called è,(with è, = ei[À~xfj)- For the Poincaré transformations (A, a) we use a 5 x 5 matrix representation (42) with the inverse /

1

0

\

(43)

The affine frame (41) can thus be obtained from the frame È = (0, e) by a boost (À, x), i.e. E = È B~\À, x). (44) Two affine frames E and È are related by a Poincaré transformation according to È = E B~\A, a)

(45)

where we have chosen B 6 180(3, 1) to act as B"1 on the right. Written in components Eq. (45) implies g, = Si [ A ~ y „ 12)

(46)

To avoid confusion with a column vector x we write x as a reminder that a Lorentz invariant object is meant.

and

Fortschr. Phys. 32 (1984) 8

461

z i = A1iZi + a i ,

> (47)

showing that the origin of the frame suffers in the tangent space Tx at x € Ui a finite displacement characterized by the vector a and a Lorentz rotation given by the matrix A. The affine frame bundle over a space-time Uit LA(Ut), is the totality of affine frames E(x) at all x £ i/ 4 which has the structure of a principal fiber bundle over i/ 4 the fiber and structural group of which is identical to the Poincaré group with an action ISO(3, 1) on itself as given by Eqs. (44) and (45) : LÀ{Ut)

= P[Ut,

F = G = IS0(3,

1)).

The affine tangent bundle over Ì74, Ta(UI), is an associated soldered bundle possessing as fiber the homogeneous space ISO(3, l)/$0(3, 1) isomorphic to Minkowski space M„ (considered as an affine space) and having the structural group G = ISO{3, 1) acting as a group of motion on the fiber : TA(Ut)

= E{Ut,

F = ISO(3,

l ) / £ 0 ( 3 , 1 )\G=

ISO(3,

1)).

An affine connection [sometimes called a "generalized affine connection" (KOBAYASHI and NOMIZU (1963)] is a connection on L A (Ut) related in a one-to-one way to a linear connection cu on L(U4) [compare LICHNEBOWICZ (1962)]. I t has the following 5 x 5 matrix form W = d

(48)

where « is a linear connection and 6 is the soldering form being a vector valued one-form which is related to the fundamental one-form 6 (see below)13). Performing a Poincaré gauge transformation ( A ( x ) , a(x)j on LA{U^), the affine connection (48) undergoes the following inhomogeneous transformation W = BWB'1

- BdB-1.

(49)

Using Eqs. (42) and (43) this implies for the entries 8 and & of W the transformation rule a> = AwA~l

(50)

— AdA'1,

and 0 = Ad + Va

with

Va = da — ma.

(51)

Eq. (50) is the familiar inhomogeneous transformation formula for a linear or Lorentz connection on L(i7 4 ), while 6 is seen to suffer a Lorentz rotation under the yl-part of the transformation and an inhomogeneous transformation under gauge translations. I t is now apparent that one can obtain the form (48) of the affine connection with the help of a boost {À, x) [see Eq. (44)] from the form

13

) 8 and S in (48) have different length dimensions (see Eqs. (55) and (56) below). The same is true for a and A in Eq. (42). 0 and a are of dimension [ i 1 ] while 35 and A are of dimension [£ 0 ]. One could introduce a length parameter in (48) and formulate the theory in terms of a connection W having a definite dimension [£ 0 ]. The translations would then have to be scaled in a corresponding manner. We, however, refrain from introducing such a length parameter at this place since the Poincaré transformations (42) automatically keep track of the different dimensions.

5*

462

W . DRECHSLER, Poincaré Gauge Theory-

yielding & = AcoA'1 -

(53)

AdA-1,

0 = Ae + Vx,-

(54)

where V denotes the covariant derivative with respect to the transformed connection &. For a pure translation (A = 1 ) this implies, writing again components = o)ik 0< = 0< +

(55) Vx1-

with

Vx1

=

dxl

(56)

+ wjx".

The soldering forms 81 are thus seen to differ from the fundamental one-forms 6' by the covariant differential of the affine vector field x l . Performing a pure gauge translation by an arbitrary finite amount (A = '\, a(x)) Eqs. (47), (51), (55), and (56) yield 6* = 6' +

+

•

(57)

One can thus comment Eqs. (55) and (56) by saying that Wik and dl define an affine connection on LA(U4) associated in a unique way with a linear connection coifc. Introducing the connection coefficients r ^ and v j by the equations mik = dxT/iik,

(58)

0* =

(59)

where [compare Eq. (3)] V

= V

+

with

Vfi =

+ rjx

k

,

(60)

one sees that F ^ = l.jF.^ [compare Eq. (31)] plays the role of a set of 24 rotational gauge potentials associated with the Lorentz part of the Poincaré transformations, while Vp = X,} + V¡¿fi plays the role of a set of 16 translational gauge potentials [DRECHSLER (1982a)]. The v j transforms inhomogeneously under gauge translations (translations in the local tangent space T x !) due to the contribution V^x1 of the affine vector field x l . We observe in passing that it is possible to transform V^x1 locally, at a given point x0 6 Uit to —dj [LICHNEROWICZ (1962)] so that Eq. (60) takes the form: •V = ik = —coik, to the result" that x'xi is a constant. Using the residual Lorentz gauge symmetry and a global translation would then transform the affine vector field everywhere to zero. One now has to focus attention on the question how matter fields transform under the gauge group ISO(3, 1) and whether their dynamics (which is later to be related to the geometry through field equations) depends on the affine vector field x(. We shall see that if we demand that matter fields transform as linear representations of the Poincaré group, the affine vector field will enter the formalism as a Goldstone-type field. If, on the other hand, the field x l is suppressed, the Poincaré gauge symmetry is spontaneously broken to the Lorentz gauge symmetry [see TSEYTLIN (1982)] and the translational gauge degrees of freedom are absent from the theory.

IV.

Poincaré Gauge Fields and Field Equations a) Generalized Matter Wave Functions

Some years ago Lur§at has suggested to use fields in particle physics which are defined on the ten parameter homogeneous space of the Poincaré group instead of the usual fields defined over Minkowski space-time [LTTRQAT (1964)]. We follow a similar line here in connection with a gauge description of gravity based on the Poincaré group [for a detailed account see DRECHSLER (1982a, 1982b)]. The idea is to describe the classical metric theory of gravitation as the Lorentz part of a gauge theory of the Poincaré group the additional translational gauge degrees of freedom of which couple via a spin-type current to the torsion of the underlying Riemann-Cartan space-time. Thus while the classical metrical field of the space-time geometry in Einstein's theory is coupled to the energy-momentum distribution of matter described in a classical macroscopic manner, the torsion induced in the Z74 geometry is related to the properties of matter represented in terms of a generalized quantum mechanical wave function . The wave function 4> is the carrier of the Poincaré gauge symmetry in a similar way as the ordinary quantum mechanical wave function in atomic physics is the carrier of the ?7(1) gauge symmetry related to the electromagnetic interaction. ist thus considered as an object [actually a section on a Jf-bundle over space-time, see below] which, speaking sloppily, possesses a phase measurable only modulo Poincaré gauge transformations. We, therefore, represent the generalized quantum mechanical aspects of matter by identifying with a section on a fiber bundle over £74 with structural group G = I SO (3, 1). The affine tangent bundle, TA(UI), and the affine frame bundle, LA(UÉ), are two examples of bundles having a coset space of the Poincaré group as fiber on which the Poincaré group acts as a group of motion : TA(U4) has fiber dimension four, while L4(C74) has fiber dimension ten. There is a whole sequence of soldered fiber bundles over space-time ranging between TA(Un) and LA(Ut) which we collectively call the soldered Jf'-bundles over space-time. They are of

464

W. DRECHSLER, Poincaré Gauge Theory

the form HciUi)

= E(Uit F e* JF = G/G';G = IS0(3, 1))

having base space Ut, and possessing as fiber a homogeneous space J f of the Poincaré group which is of the form Jf = X S, i.e. contains Minkowski space-time. 14 ) The soldering to the base is made through this subspace of Jf? by identifying the local tangent space of Ut at x with the Minkowski subspace of through an isomorphism [compare EHRESMAN (1950)]. The classification of the homogeneous spaces J f is obtained by utilizing a corresponding classification of stability subgroups G' of the Poincaré group. In fact, it was shown [BACRY and KIHLBERG (1969), KIHLBERG ( 1 9 7 0 ) ] that G' must be a subgroup of the Lorentz group if 2/C = M4 X S is required to hold true. The dimension of je is given by N = 10 - dim G' and ranges from N = 4 {G' = SO{3, 1)) to N = 10 (G' = 1). In these two extreme cases the bundle Ha>(U4) is identical with TA(Ut) and La(Ui), respectively, as mentioned above. We shall not specify here the particular Ji?-bundle with fiber dimension N (4 5S N Si 10) to be used in physics ; however, we will assume in the following discussion that the space J f possesses a Poincaré invariant measure and that it can carry half integer spin fields. The first requirement is necessary for the definition of a translationally gauge invariant current to be possible; the second requirement forces N to be bigger than seven [FINKELSTEIN (1955), BACRY and KIHLBERG (1969)].

It is now straightforward to define a scalar wave function (x; X) as a cross section on HGiUt) and introduce a Poincaré gauge covariant derivative for it. We call (x; X) with x e TJi and X e Jifx = Tx X Sx ^ Mt X S a Poincaré gauge field: {x;x,y);

with

xeTx-,yeS.

(66)

Here X — (x, y) denote the additional N internal coordinates varying in the local fiber i = 0, 1, 2, 3, since they x over the point x. Four of these coordinates are called x \i are identical with the affine vector field discussed in the previous section. Performing a Poincaré gauge transformation on (f>(x; X), i.e. changing the cross section on the soldered -bundle, changes the internal coordinates according to $(x;X)

= 4>(x;

,

(67)

where X = g~xX with g e ISO(ì, 1) is a short-hand notation for x = A'1(x) (x — a(x)), being the transformation in the subspace Tx of and y = y{y), being the transformation in the subspace Sx of Jf?x. [The transformation in Sx is only related to the Lorentz part of g = [A(x), a(x)), not to the translations]. Eq. (67) shows that (x; X) is a linearly transforming scalar Poincaré gauge field. A Poincaré gauge covariant derivative of 4>(x\ X) — which we denote by D ; i.e. D = 6kDk = dx^Dp — is defined in the following way15) I)(x; X) = [ci + ir] (x; X) d =+ ^r lOijMV - frPi (x;X), Ù

(68)

r = -

(69)

where tbijM* -

frPi

is the Lie algebra valued one-form associated with the Poincaré group with r defining a connection on J?G'(C/4). Here Mt.¡ and P ¡ form a representation of the Lie algebra of the 14

) In fact, J? can itself be regarded as a trivializable fiber bundle with base Mi = IS0(Z,

1)/

SO(3, 1) and fiber S = SO(3, 1 )/(?'.

15

) d = 0k 8k = dxfd^; compare also Eqs. (58) and (59). One can without loss of generality replace, moreover, a>ik by coik (see Eq. (55)).

Fortschr. Phys. 32 (1984) 8

465

Poincaré group in terms of differential operators to be applied to scalar functions defined on the homogeneous space J f :

(TO)

Pi = ih, Mij = Lij + Sij = i{Xi dj -

ócj di) + Sij.

(71)

Pi and La form a representation of the Lie-algebra of ISO(3, 1) in Minkowski space, while the Sij are differential operators in the additional variables y needed to describe the part S of the s p a c e d . [Explicit formulae for the 8¿,- were given by B a c b y and K i h l b e k g (1969)]. The S.^ behave like spin operators obeying the commutation relations Ski] = VikSjt + VjiSik — mfiik — VikSn, [ L i h S k l } = 0,

(72)

[Pi, Ski] = 0Clearly P i and M t .j satisfy the commutation relations for the generators of the Poincaré group i[Mij, Mkl] = rjucMji + ihlMik - rjuMjic v,kMu, i\Pi, Mik] [P„P,]

= rlikPj -

rH,Pk,

(73)

= o.

Using the commutation relations (73) it is easy to show from Eq. (69) that the Poinca)réLie-algebra-valued curvature form on //G'(t/4) is given by 16 ) ¿7 = df + J [r;T]=

J QijM*

-

T'Pj = 1 Qifiti -

Q'P,,

(74)

where the last equality follows from Eq. (62). From Eq. (74) one concludes that defining the wave function 4>(x ; X ) on the affine tangent bundle, Tyl(?74), having a fiber dimension N = 4, i.e. disregarding the additional ¿/-coordinates, would lead to a curvature from E given by —Q'Pj, with i2> being Cartan's torsion two-forms defined in Eq. (34). W e shall, however, see below that the operators S^ are essential in defining a bilinear source current in the fields (x; X). W e remark in closing this subsection that in order to fix a particular mass and spin value for the physical object described by the scalar field (x; X) one would have to demand that the Casimir operators of the Poincaré group associated with the space J f take definite values when applied to (x; X), i.e. that PiP^ix WiW^ix;

; X) = m24>(x; X), X) = —m2s(s + 1) {x; X),

(75) (76)

where W< = 1 e^'M,kPl

= j

E»"SikPt

(77)

is the Pauli-Lubanski operator associated with the space J f . 16)

For a Lie-algebra valued form F = FATA with fA being a set of. forms and TA a basis of the Lie algebra [T, 71] is defined by [J1, 71] = rA A rB[TATB - TBTA],

466

W. DRECHSLER, Poincaré Gauge Theory

b) Field Equations in a U4 Now the problem arises of relating the dynamics of a Poincaré gauge field (x; X) to the geometry of the underlying Riemann-Cartan space-time in such a way that the usual metric formulation of classical general relativity is one aspect of the description arising from macroscopic matter distributed in classical form, while the quantum mechanical i.e. generalized wave function aspect of the discription of matter induces torsion in the underlying space-time geometry. There is, however, one more restriction to be made in this context. The torsion tensor Sijk = —Sjiic has in general 24 independent components separating into three irreducible parts: a tensor (16 components), a vector (4 components), and an axial vector (4 components). In the following we shall assume that only axial vector torsion is present in the Riemann-Cartan space-time, i.e. we only consider one irreducible component of torsion namely the axial vector part given by a totally antisymmetric S{jk = S[i,ky In this case Kijk = ( 1 /2)(S',and the geodesies of a U} coincide with those of a V4. Now two sets of field equations are considered in a £/,, with axial vector torsion in the presence of matter represented in the form of a Poincaré gauge field (%; X) : Rik-^riikR and

= xTik,

(78)

17'Rijkl=xJkji.

(79)

Eq. (78) is postulated to contain Einstein's equations (1) for the metric, while Eq. (79) turns out to be a set of independent nonlinear field equations coupling what will be called a translational spin current bilinear in (x; X) to the axial vector torsion field. The coupling constant x in Eq. (79) is regarded to be different from x. Eq. (79) may thus be interpreted as a geometrically motivated field equation determining an interaction which may be different from gravitation although it appears here in a geometric framework together with the gravitational interaction. We now decompose the source terms l \ k and J k j i = —J k i j into a classical part describing macroscopic matter, and a (x\ X) representing the quantum mechanical aspect of matter. Exhibiting explicitely the symmetry properties of the various parts we write17) Tik=Tiik)

+ Tik(4>),

(80)

xJk[ji] = x-hm + *J[kH] W>)-

(81)

Here Tik = Tki is the classical symmetrical energy-momentum tensor which is the source term in Einstein's equations, and Tik() + Tiik](4>) represents the (^-contribution to the total energy-momentum tensor. With Eqs. (80) and (24) Eqs. (78) yield xTm() =Piik)--ilikP, y-Tm{4>) = pm 17

= -

(82) '

(83)

) Round brackets indicate a symmetric pair of indices; square brackets indicate an antisymmetric set of indices.

Fortschr. Phys. 32 (1984) 8 with

467

" Pit

+ P[ik] = Piikrf1;

= Pm

and

P =

Pikr?k.

where P^M is given b y E q . (38) (here for totally antisymmetric Kijk). T h e last equality in E q . (83) follows f r o m the contracted Bianchi identities (39') with the torsion-torsion part being absent on the right-hand side of this equation in the case of axial vector torsion. Eqs. (82) and (83) will be regarded as identities defining the quantities Tlk(4>) in terms of the P i k . Turning now to E q . (81) we observe that the classical contribution, Jkji, to the current, obeying J{¿¿¿j = 0 , is a derived quantity in general relativity which thus need not be defined separately. I n fact, the contracted Bianchi identities in a V i yield VlRm =

V % k j i = VfSik

-

ViR ]k = : xJkji,

(85)

showing that the J k j i are expressible, with_the help of the field equations (24), in terms of metric covariant derivatives of Tik and T = Tlkrjlk. The influence which matter described in quantum mechanical, i.e. generalized wave function?form has on the underlying Riemann-Cartan geometry is expressed through the current Jkji() — supposed to be totally antisymmetric in all its indices. This part of the current induces an axial vector torsion field in the geometry as we shall see. I n fact, using Eqs. (38), (81) and (85), and contracting the resulting equation, on the one hand, with the Levi-Civitta tensor esiik and, on the other hand, with i f k and using the contracted Bianchi identities as well as the complete antisymmetry of the current Jkji((j>) as well as the torsion field Kkli, one obtains f r o m E q . (79) the following t w o sets of field equations [DRECHSLER (1982a, 1982b)]. V'*Pit V'Pu

= =

Vl*Pu VlPn

-

Kj«*Pm

K j « P

m

(86)

= -Z**Ji() = ~^esiikJijM)>

(88)

*KS = ~jesi'kKllk,

(89)

*Pilik=^eijHPpqlk,

(90)

*Pii Moreover, Pit

= 2[Vt*Ki -

= *Pijikrfk

(91)

r)i,Vs*Ks]•

can be expressed in terms of *KS in the following w a y

• P « = Pm

+ P

w

= eu'%*Kk

+

2[*Ki*K, -

Vi,*Ks*Ks].

(92)

W e , finally, write the equations (86) and (87) as nonlinear equations for the axial vector *KS and obtain P%*K, -

j

di(*K°*Ks)

ds(^*K,) -

= j

e^KiV+K, -

*Ks(Vs*Ki)

+ j

*Ki(Vs*Ks).

x*J,{),

(93) (94)

468

W. DRBCHSLER, Poincaré Gauge Theory

For a study of the inhomogeneous field equations (93) — called the current-torsionequations — and the nonlinear constraint relations (94) in conjunction with the topological invariants of the space J74 see DRECHSLER ( 1 9 8 2 b). _ We point out in passing that the i/4-curvature scalar is B = R + P, and that *P = *Pikrfk with the quantities P and *P given by P = -Q*KS*KS,

(95)

*P = -6Va*Ks.

(96)

and Moreover, Eq. (93) implies the following divergence relation for the axial vector current *Ji() [compare DRECHSLER (1982a)] ^*Ji(4>) = V**Jm

= ^ {eWtyfKj)

(Vk*Kt) + PitfVtK,)}.

(97)

This relation shows that the axial current *«/;($) possesses an anomaly which in a metrically flat background space (i.e. for V{ — 3; ; R i f = 0) has a form similar to the anomaly of the axial vector current in spinor electrodynamics [ADLER ( 1 9 6 9 ) , see also KIMURA (1969) for an extension to general relativity]. Let us now define a totally antisymmetric source current Jijk{4>) i n terms of the Poincaré gauge field (x; X). We obtain it in a unique way by integrating over the local fiber a totally antisymmetric bilinear density in the -fields constructed with the help of the generators (70) and (71) of the Poincaré group associated with the homogeneous space ¿4? : Jijk(4>) = jjTF f

*(x;X)M{ijPk}(x-,X)d^X).

(98)

X

Here d/i(X) denotes the Poincaré invariant measure on Jti?, and R 0 is a fundamental length parameter18) normalizing the current to give it the correct length dimension [£~ 3 ] required for Jijk() to act as a source current in Eq. (79), if x is considered to be a dimensionless coupling constant and the 0-field to have the canonical dimension [Z^1]. The power N with which R0 appears in the denominator on the right-hand side of Eq. (89) is equal to the fiber dimension of the space Hg^V,). N thus effectively determines the strength of the coupling in the current-torsion-equations. The integral over the local fiber in eq. (98) yields a quantity defined on the base which, indeed, can be used as a local source current. Remembering Eqs. (70) and (71) and observing that L{ijPk) = 0, we see that the dual current *Ji() is given by

*Ji{4>)

=

~ w f

je*

4 > * { x ' 1 ] Wi{x;

d A l )

(99)

with being the Pauli-Lubanski operator associated with the homogeneous space x which was defined in Eq. (77). We thus see that the spin-type operators 8 as well as the translation operators Pk enter the definition of the source current for the torsion field which justifies calling *Jl(4>) a translational spin current. The form of the current (99) implies, moreover, that the spin degrees of freedom are essential and that it is not possible to base the theory on the affine tangent bundle, 7,j4(f/4), possessing the smallest possible fiber dimension N = 4. 18)

The fundamental length parameter R 0 will basically be introduced into the theory through a spontaneous symmetry breaking described in subset c. Here it appears as a dimensional constant appropriately scaling the matter source current.

Fortschr. Phys. 32 (1984) 8

469

Using Eq. (67) and the Poincaré invariance of the measure the index i of *Ji{) is seen to transform as a local Lorentz index under Poincaré gauge transformations as is required for Eq. (86) to be Lorentz gauge invariant. Of course, both sides of Eq. (86) are axial vectors, i.e. behave under parity transformations in Tx as *J"() -> —*J°(4>) ; *J() —> *J() and analogously for *K\ Before we turn to the equations for 4>(x; X) we remark that in the absence of a 0-field one recovers Einstein's metric theory of gravitation coupling the metric to a classical source distribution of energy and momentum. In fact, in the absence of matter to be treated in a quantum mechanical manner one would have to pose the conditions Tik(4>) = 0, and *Ji() = 0. In this case the equations (82) and (83) together with (86), (87) and (91) lead to the conclusion that *KS must vanish identically, implying that there is no torsion disconnected from, the material sources. Thus torsion is not possible in this theory as a vacuum phenomenon; if the (x; X) to hold true on the J f bundle: Dk{%; X) = bk4>{x; X). (100) The field bk introduced here has dimension and would have to transform only under the Lorentz part of the Poincaré transformations in order for Eq. (100) to be a ISO{3, 1) gauge invariant equation. We, however, intend to break the Poincaré symmetry [without eliminating the fields x1 from the geometry] by taking bk proportional to the affine vector field xk transforming according to Eq. (47) and put Dk4>{x-X) = ^{x-X).

(101)

Here R 0 is the fundamental length parameter introduced before. Eq. (101) can be viewed as a Poincaré gauge fixing condition freezing the translational part of the Poincaré gauge symmetry and leaving the Lorentz part untouched. The condition (101) restricts the so far unspecified dependence of 4>(x; x, y) on xk ; let us see what the consequences are. We first observe in contracting Eq. (101) with xk that for \xkxk\ R02 one finds that xkDk 0, implying that c/> is essentially parallel shifted on the ^ - b u n d l e over distances which are small compared to R 0 . To have a better understanding which operators of the Lie algebra of the Poincaré group are actually involved on the left-hand side of Eq. (101), let ns rewrite this equation using Eqs. (56), (68), (70) and (71) : 0* + ( V + 19

Si +

Y

(x-,x,y) = —{x;x,y).

) In the following discussion we deviate from the arguments presented in

DRECHSLER

(102) (1982a).

470

W. DRECHSLER, Poincaré Gauge Theory

Differentiating Eq. (101) a second time, and considering the fact that the first Poincaré covariant derivative has produced a Lorentz vector index, one obtains [denoting the generally covariant differentiation by D; and suppressing the arguments of DiDó

= (DiDk - riksDs)

= ^r

(103)

^0)'t>'

+

where we have used Eq. (101) again on the right-hand side. Taking the symmetric and antisymmetric components in i and k of this equation one has j

(DiDk + Dtfii) * =

j(Vixk+Vkxi)+^-0

~

1 {DiDk - DkDi) = - L [ViXk -

(104)

4>,

(105)

FkXi] 4>.

Contraction of Eq. (104) with r\ik yields Vi&

' R