Quantum Computing and Writing: Exotic Quantum Statistics and the Translation of Quantum Texts
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English Pages [177] Year 2005
1 GENERALIZED STATISTICS OF INDISTIGUISHABLE PARTI-
CLES 1
1.1 The Symmetrization Postulate 1
1.2 Identicality and Indistinguishability 2
1.3 The Quantum Statistics of Indistinguishable Particles 5
1.4 Cluster Decomposition: Systems and Subsystems 7
1.5 Conclusion 8
2 STATISTICAL MECHANICS OF IDENTICAL PARTICLES 9
2.1 Multiparticle States and the Permutation Group 9
2.1.1 Notation and Fimdamentals 10
2.1.2 Frobenius Symbols 10
2.2 Partition Fvmctions for (In)distinguishable Particles 12
2.2.1 Parabosons and Parafermions 19
2.2.2 Beyond Parabosons and Parafermions 28
2.3 Degenerate Systems 35
2.4 Conclusion 37
3 STATISTICAL MECHANICS OF EXTENSIVE SYSTEMS 39
3.1 Extensivity 39
3.2 Extensivity and Positive Counting 41
3.3 Classification 45
3.4 Symmetry Types in Fuzzy Statistics 47
3.5 Examples and Comparisons 50
3.5.1 Interpolating Between Bosons and Fermions 50
3.5.2 Haldane/Wu and Polychronakos's Fractional Statistics 52
3.5.3 Gentile Statistics 54
3.6 Conclusion 56
4 SECOND QUANTIZATION AND GENERALIZED STATISTICS 57
4.1 Algebras: Bilinear and Trilinear Relations 58
4.2 Actions of the Permutation Groups 63
4.2.1 Gram Matrices and the Representations of the Permutation
Group 67
4.2.2 Clustering Revisited 81
4.3 Classification 85
4.3.1 Examples 90
4.4 Known Models 95
4.4.1 Green's Parastatistics 95
4.4.2 Greenberg's Quons 96
4.4.3 Meljanac's proposal and Govorkov's Parastatistics 96
4.5 Conclusion 98
5 ENSCRIPTION AND TRANSLATION OF QUANTUM TEXTS . 100
5.1 Cloning and Quantum Information 100
5.2 Enscription 104
5.3 Enscribing 2-texts 113
5.4 Enscribing Real-Uniform N-texts 116
5.4.1 Real-Uniform 3-Texts 119
5.4.2 Real-Uniform N-Texts 128
5.5 The Structure of an Enscribable Text 133
5.6 Enscription as a Resource: Probabilistic Clorung 135
5.7 Translation 142
5.8 Graphs and the Structure of Translatable Texts 145
CITED LITERATURE 162
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QUANTUM COUNTING AND WRITING: EXOTIC QUANTUM STATISTICS AND THE TRANSLATION OF QUANTUM TEXTS
BY RANDALL ESPINOZA B.S. Universidad de Costa Rica 1994
THESIS Submitted in partial fulfillment of the requirements for the degree of Doctor of Philosophy in Physics in the Graduate College of the University of Illinois at Chicago, 2005
Chicago, Illinois
UMI Number: 3199854
INFORMATION TO USERS
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THE UNIVERSITY OF ILLINOIS AT CHICAGO Graduate College CERTIFICATE OF APPROVAL
1 hereby recommend that the thesis prepared under my supervision by RANDALL ESPINOZA
entitled
QUANTUM COUNTING AND WRITING: EXOTIC QUANTUM STATISTICS AND THE TRANSLATION OF QUANTUM TEXTS
be accepted in partial fulfillment of the requirements for the degree of DOCTOR OF PHILOSOPHY
cr-yy1 is denoted by J and
labeled by the arrays {jr+i, ••• ,jM), with ;V+i < • • • < /m- Expanding the determinant in the numerator of
(x, y) we have
.v;x)=^E='8nw ^ > O-
n C"' ,V
n i=jr+1,-,iM
22
It should be clear from Equation 2.21 that function for the partition
. .,V;x) equals, up to a sign, the Schur
defined by the array
(I) _ \ M + q+i k - l , i k i n l ,
' M - jk,
h > • • • > ir)
jk in J ,
jr+i < • • • < ; m -
The partition ?J clearly has a diagonal of length r (which is how many
are bigger than
M) and is best represented by its Frobenius symbol
Fr(Ai) = [ q + h - l , . . . , q + i r - l \ h - l , . . . , i r - l ] .
(2.36)
(z'L, ..., zV; x) and the Schur function S AJ (X ) is the result of the
The relative sign between
permutation of the lower indices in Equation 2.34 for
{h,---, ir'> x) relative to the order of
the columns in the determinant in Equation 2.21 — that is, it is the sign of the permutation
and is given by
H
^2
• ••
V
1
2 ...
r
]r+l
• • •
r + 1 ...
]M
\
M )'
'
Using the form of Fr(Ai), it follows that
E(^'-l) = |bAzllei We conclude that
/n oyx
^
(2-38)
..., zV) = (-1)I'''^ISAJ (X ), and more generally that/(°'^^(x,y) is given
by det(x^-' + vx''.^ ' , M-,x
)
)i 0, or of the form (i, i + |sl) — the diagonal starting at the th box of its first column — for s < 0. We denote by rA(s) the length of the s-diagonal of A. It follows that for a partition A = (Ai,..., Am), the length of the s-diagonal rx{s) is given by the number of parts Ayt that satisfy
> k + s. Equivalently, in terms of the associated array
i = {l\,.. .,£m) introduced in Equation 211, the s-diagonal is the number of 4's for which 4>M
-I-
s. Figure 2.2.2 clarifies this concept with some examples.
33
Fr(X) = [3,1,014,2,1]
Fr(\) = [4,2/ll3,1,0]
1-diagonal (- I )-diagonaI '-x(-l) = 3 Figure 2, Two examples of the concept of the s-diagonal.
Defiiie the function ^s(0 on the ring of formal symmetric fimctions as
(2.81) A
Since ^-s(0 = ^s(0/ we assume without loss of generality that s > 0. When dealing with the CGPF of systems with (p,£^)-statistics, the parameter s will usually stand for p - q. For instance, for p - q > 0 the (p, (j)-envelope
introduced before can be equivalently defined
as the set of all those partitions A for which r,\ (p - q) < q. With this in mind it follows that Lemma 2.7. f4^ = EZ(,(2-82)
34
Proof. The left hand side of Equation 2.82 is
( £ i") A
/
\n=0
/
A
(2.83)
«=rA(s)
00
'?=0
E «a(x )), A,rA(s)