Gauge Field Theory in Natural Geometric Language: A Revisitation of Mathematical Notions of Quantum Physics 9780198861492, 0198861494

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Gauge Field Theory in Natural Geometric Language

Gauge Field Theory in Natural Geometric Language A revisitation of mathematical notions of quantum physics

DANIEL CANARUTTO

1

3 Great Clarendon Street, Oxford, OX2 6DP, United Kingdom Oxford University Press is a department of the University of Oxford. It furthers the University’s objective of excellence in research, scholarship, and education by publishing worldwide. Oxford is a registered trade mark of Oxford University Press in the UK and in certain other countries © Daniel Canarutto 2020 The moral rights of the author have been asserted First Edition published in 2020 Impression: 1 All rights reserved. No part of this publication may be reproduced, stored in a retrieval system, or transmitted, in any form or by any means, without the prior permission in writing of Oxford University Press, or as expressly permitted by law, by licence or under terms agreed with the appropriate reprographics rights organization. Enquiries concerning reproduction outside the scope of the above should be sent to the Rights Department, Oxford University Press, at the address above You must not circulate this work in any other form and you must impose this same condition on any acquirer Published in the United States of America by Oxford University Press 198 Madison Avenue, New York, NY 10016, United States of America British Library Cataloguing in Publication Data Data available Library of Congress Control Number: 2020945386 ISBN 978–0–19–886149–2 Printed and bound by CPI Group (UK) Ltd, Croydon, CR0 4YY Links to third party websites are provided by Oxford in good faith and for information only. Oxford disclaims any responsibility for the materials contained in any third party website referenced in this work.

Rn

Cn

BV V AutV ⊂ EndV

V

BV

G ⊂ AutV

Rn

EndV ∼ = V ⊗V ∗

V

L ⊂ EndV L

E!M E → JE M

JE ! E ! M

M JE ! E

E!M

LCE ! M T∗M ⊗ EndE ≡ T∗M ⊗ E ⊗ E ∗ ! M . M

M

M

E!M

L ⊂ EndE

K ⊂ LCE K!M

T∗M ⊗M L ! M

κ0

M →K

κ κ0

κ − κ0

S!M

H!M S

• • •

•

•

x∈M f (x1 , x2 , .., xm )

M

x : M → Rm f

m f

Tr prj

j

s C : V → V : v (→ v¯ ex : W →W ⋆ T V J ≡ J1 Jk

k

j ≡ j1 jk

k

D d

ˇ dH , dV , d L[v] δ[v] S

v (→ v(k)

k

∗

∇

∇·

d[κ], d· 2

∇ ∇ " ∇(n) ˇ ∇ D

D, d F+ F− ˆ φˇ φ (→ φ, a, a∗ op φ (→ ⌊φ⌉ e i sgn H Em (p)

H

≡ 12 (sgn + 1) p

C, P, T dl, ν rk ϕ

k

∗

ϕ

ϕ∗

ϕ

v (→ v

♭

λ (→ λ#

j, J

α, π, ι

E ⊗B E ′

⊗

⟨λ, v⟩

⟨ θ, u⟩⟩

¯ ∧, ∨, ∨

♦,

Z2

♦

τ ⌋t τ |t

Z2

¯ ∧ †

[X, Y ] G (X, Y

X

Y

)

[u, v]

u

v

[[ζ, ξ]] {[X, Y ]}

∆, { , }

V ,W ...

ζ X

Y

ξ

V →W

Lin(V , W )

V →W

Lin(V , W ) EndV ≡ Lin(V , V )

V

l : L )→ EndV V ∗, V ⋆ V ,V ⋆ M, N, X . . . M p:F !B

E!B Z!B ZN ! N ⊂ X

Z!X Z!X

Zx

x∈X

E = E⌊0⌉ ⊕M E⌊1⌉ P ≡ T∗M

Pm , Pm− ⊂ P P (r) ! M ∥

P =P ⊕P

M r

⊥

T ⊂M

t:M !T DA ! B

A!B

LCE ! B

E!B

K ⊂ LCE S

M

U

M

Q ≡ ∧2 U

¯ U⋆ H≡U∨ N, N

±

U

⊂H

W ≡ U ⊕ U⋆ W± ⊂ W

U ≡ EndU

H ≡ EndH G⊂H

H

g

J

Ωr ≡ ∧r T∗M

U, S , V

L, M, T

r

M

V ≡ VX ! X

X

L, BL , BLC ER , EL

M

LR , LL

ER

Y ≡ YR ⊕ YL

M

YR ≡ ER ⊗ U

Y

YL ≡ EL ⊗ U ⋆

Y

I!M

E ≡ EndI

E′ , E′′ , E+ , E− ˜ (h,k), W {r} U (h,k), U

E

∆ ⊂ X×X

D D◦ DV D D◦

D◦ ⊂ H ⊂ D S, S◦ O

E ⊂ O⊗E

E!M

˜ E

E = E⌊0⌉ ⊕M E⌊1⌉ ˜ =E ˜⌊0⌉ ⊕M E ˜⌊1⌉ E Z n ≡ ♦n Z 1 ! n Z≡ ∞ n=0 Z

n

Z⋆

V ≡ Z ′ ⊗ Z ′′ ⊗ ··· V♦ ∼ = V ⊗ V⋆ D, E, V, Z, O ℘:Z!M (℘, Y):ZX →X×Y Q

Z!P !M X⊂M Z!P !M

EL

11 11E ←

θ | θ| θ ˘ θ ˘[2] θ

g g# |g| g⋄ d d η η⊥ p (→ (p∥ , p⊥ )

h k γ γη χ ≡ ±iγη γ(n) κ ρ ∗

∗

κ, ρ νκ

κ ∗

¯, κ ¯ κ

κ ⊗ κ′ κ0 A

≡ κ − κ0

F Φ

b, β f, ϕ

gL , ηL , ∗L E, B

Γ T T˘ = T bab dxa

R Ric ⟨R⟩

E Pm ! M

ˇ Γm , Γ Γ Ω ˜ Γ′ Γ

V∗Pm ! M

−

G Y

Γ ΩD

− D

∆Γ, ∆θ

L ℓ

L = ℓ dx

E

P, C ≡ P+L J , U, T

K, M, N ≡ SM Λ, λ

Π(0), Π(1), Π(2) Υ H, P L, H P φ, φ¯ ψ, ψ¯ ψ † ≡ ψ¯ γ0 ω, ϖ, N f

hR , hL

ER

EL

Ψ ≡ (ΨR , ΨL ) Ψ ≡ (ψ , ν) φ; H, H0

φ+ , φ− , φ0 W , A, B , Z , W

±

K(p) Φ I, J , M, S, S ′

W →W

p ∈ Pm

σ:X$Z

Z!X {x}

δ[x] ω[f]

f

pv(θ/f)

δ∆ ω± m ± α± m , βm , om [±±] − em , e+ m , em ± ◦ • Dm , Dm , Dm , Dm , Dm , Dm

D± , D◦ , D• , D , D , D D± , D◦ , D• , D , D , D ± ◦ • Dm , Dm , Dm , Dm , Dm , Dm

" " κ, ρ Q

κ

∈O

a, b, c, g, k

C, C H=iK U, S %KN & ˘, Q, Q, Q

# a$ x

dx ≡ dm x # a i$ x ,y # a a$ x , x˙ # $ # $ pλ ≡ p0 , pi |A|

¯α˙ bα b ¯α˙ bα b

δαβ # $ zA ⊂ U

KN (t1 , . . . , tN ) ≡ K(t1 ) ◦ · · · ◦ K(tN )

dx1 ∧ .. ∧ dxm

T∗M A

# #

$ # $ ζα , ζ′α ⊂ W $ # $ ζα ≡ u A ; v B

W±

ϵ ∈ ∧2 U ⋆ ϵ♭ , ϵ #

ϵ

σλ , σλ # $ # λ$ τλ τ ς ∈ ∧2 I ⋆ # $ lI

cIJH # a α I α I $ x , y , ka ; ya , ka,b

#

xa, yα, kaI; zaα, zabI

da f # $ # $ ξα , ιµ # ′ ′′ + − $ e ,e ,e ,e % & xα & Bxα , {B

JF

$

DF dH f

E ≡ EndI

I

E

z xα , ζxα

axα , a∗xα Xp

, Bpα , Apα , CpA˙

y⌊i0⌉ , y⌊i1⌉ →

←

∂i , ∂i ← ∂⃗ i , ∂ i ∼

∼

qxi , Πxi

c, G, ! θ ll ∈ L

q, e ∈ R ξ

θ ∈ O⌊1⌉

µ ∈ Lr−4

r

.

.

(X, x)

. . .

.

.

m

M x:X →O ⊂ A O % & A M (Xα , xα ) Xαβ ≡ Xα ∩ Xβ ̸= ∅

X⊂M

m

←

xα ◦ x β : xβ (Xαβ ) → xα (Xαβ ) A x:M →A

Gauge Field Theory in Natural Geometric Language: A revisitation of mathematical notions of quantum physics. Daniel Canarutto, Oxford University Press (2020). © Daniel Canarutto. DOI: 10.1093/oso/9780198861492.001.0001

y:N →B

N f :M →N

y◦f

◦

←

x :A→B.

f

# $ x ≡ x a : M → Rm , f :M →N f

#

f i (xa )

$

# i$ y : N → Rn , f i ≡ yi ◦ f : M → R

n

F

B

p:F !B ← Fx ≡ p (x) ⊂ F x∈B Y ⊂F X ≡ p(Y ) ⊂ B

ϕ : F → F′

B n ≡ dim F − dim B (xa , yi ) : Y → Rm × Rn a x Yx x ∈ X # a$ # a ←$ x ≡ x ◦ p : X → Rm B p:F !B p′ : F ′ ! B ′ ϕ : B → B′ p′ ◦ ϕ = ϕ ◦ p p:F !B s:B→F p ◦ s = 11B F !B F′ ! B F ×B F ′ ! B ←

Y ≡ p (X) ⊂ F

E!B A!B

X⊂B

E∗ ! B

DA ! B B→A E!B E′ ! B

A!B

DA ! B

E ⊕ E′ ! B ,

E ⊗ E′ ! B ,

B

B

E ⊗ E′ ! B EndE ≡ E ⊗ E ∗ ! B E!B 11 ≡ 11E : B → EndE

∨rE ⊂ ⊗ rE ! B

∧r E ⊂ ⊗ rE ! B r

∧E ≡

yi E!B

n !

r=0

∧r E ! B

# i$ y

m c(t0 ) = c˜(t0 ) = x

#

M t0 ∈ R

t0 ∂c(t0 ) ≡ ∂˜ c(t0 ) TM ≡ M TM ! M

$ xa , y i E∗ ! B

'

Tx M ! M # a$ x M d x˙ a ◦ ∂c = c˙a ≡ dt (xa ◦ c) M x∈M

# $ yi

c, c˜ : R → M Tx M

m

#

xa , x˙ a

$

T∗M ! M f :M →R d ⟨df ◦ c , ∂c⟩ = dt (f ◦ c) df = ∂af dxa

∂af ≡

f

∂f ∂xa

TM ! M df : M → T∗M c:R→M # a$ xa dx T∗M ! M a x˙ ← f ◦ x : Rm → R

T∗M x˙ a ← t (→ x (x1 , ... , xa + t, ... , xm ) N Tϕ : TM → TN # i$ c:R→M y

ϕ

∂a ≡ ∂xa # $ ∂xa

#

#

xa , x˙ a

dxa

$

$ ϕ:M →N Tϕ ◦ ∂c = ∂(ϕ ◦ c)

N

(yi , y˙ i ) ◦ Tϕ = (ϕi , ∂a ϕi x˙ a ) . T(ϑ ◦ ϕ) = Tϑ ◦ Tϕ

T β : N → T∗N

ϕ∗β : M → T∗M ⟨ϕ∗β, v⟩ = ⟨β, Tϕ(v)⟩ ,

v : M → TM .

ϕ∗ (β ⊗ β ′ ) = (ϕ∗β) ⊗ (ϕ∗β ′ ) τ : N → ⊗ T∗N u : M → TM v : N → TN Tϕ ◦ u = v ◦ ϕ ϕ∗ (v ⌋τ ) = u⌋(ϕ∗ τ ) , ϕ

⌋

M

u : M → TM

ϕ ←

ϕ∗ u ≡ Tϕ ◦ u ◦ ϕ : N → TN . f :M →R

Tf : TM → TR ∼ =R×R

df ≡ pr2 ◦ Tf : TM → R . +1 M → ∧T∗M

d(df ) = 0 d(λ ∧ µ) = dλ ∧ µ + (−1)r λ ∧ dµ ,

df

λ : M → ∧r T∗M µ : M → ∧s T∗M

λ ∧ µ : M → ∧r+s T∗M

λ : M → ∧r T∗M λ = λa1 ...ar ∧ dxa1 ∧ · · · ∧ dxar ,

dλ : M → ∧r+1 T∗M

λa1 ...ar : M → R ,

dλ = dλa1 ...ar ∧ dxa1 ∧ · · · ∧ dxar = ∂b λa1 ...ar dxb ∧ dxa1 ∧ · · · ∧ dxar . λ θ

dλ = 0 d2 λ ≡ d(dλ) = 0

λ = dθ ϕ ϕ∗ (λ ∧ µ) = (ϕ∗ λ) ∧ (ϕ∗ µ) , r−1

ϕ∗ (dλ) = d(ϕ∗ λ) ,

u : M → TM v : N → TN v r

v|λ

u : M → TM

v ∈ Tφ(t,x) M α ∈

φx (t)) = ⟨df, u⟩ ◦ φx (t) ,

t (φt )∗ α ∈ T∗x M .

v : M → TM

α : M → T∗M

[u, v] ≡ L[u]v ≡ lim

t→0

(φ−t )∗ v − v : M → TM , t

L[u]α ≡ lim

(φt )∗ λ − λ : M → T∗M . t

t→0

v|λ = r v ⌋λ −1 • (R × {x}) ∩ X • φ(0, x) = x

{0} × M x∈X

• φ(t′ , φ(t, x)) = φ(t′ + t, x)

u

f :M →R.

∗ Tφ(t,x) M

(φ−t )∗ v ∈ Tx M ,

X

Tϕ ◦ u = v ◦ ϕ iv λ

λ

φ:R×M →M φx : R → M : t (→ φ(t, x)

x∈M d (f ◦ dt

ϕ∗ (v|λ) = u|ϕ∗ λ ,

λ !→ v|λ M

R×M

x∈X

R×M

X

u

v

•

L[u]f = u.f ≡ ⟨df, u⟩

f

• L[u]v = [u, v]

u

v

[u, [v, w]] + [v, [w, u]] + [w, [u, v]] = 0 , M

u = ua ∂xa ,

v = v a ∂xa , α = αa dxa ,

τ = τ ab...cd... ∂xa ⊗ ∂xb ⊗ · · · ⊗ dxc ⊗ dxd ⊗ · · · ,

u.f = ua ∂a f , [u, v]a = (ub ∂b v a − v b ∂b ua ) , (L[u]α)a = ub ∂b αa + αb ∂a ub , (L[u]τ )ab...cd... = ue ∂e τ ab...cd... − τ eb...cd... ∂e ua − τ ae...cd... ∂e ub

+ τ ab...ed... ∂c ue + τ ab...ce... ∂d ue + · · ·

L[u]df = d(u.f ) ,

L[u]λ = d(u|λ) + u|dλ

ϕ:M →N Tϕ ◦ u = v ◦ ϕ ϕ∗ (L[v]µ) = L[u](ϕ∗µ) ,

TF

#

xa , y i

$

F TF p:F !B #

u : M → TM v : N → TN µ : N → ∧T∗N .

a = 1, . . . , m i = 1, . . . , n # a $ dx , dyi T∗F #

xa , x˙ a

$

◦

(λ : M → ∧T∗M ) .

# $ Tp = xa , x˙ a

p:F !B #

∂xa , ∂yi

$ xa , yi ; x˙ a , y˙ i TF Tp : TF ! TB

$

T∗B → T∗F

p

Tp T∗F ! F

r∈R ∧r T∗B ⊂ ∧r T∗F F → ∧r T∗B ⊂ ∧r T∗F VF ⊂ TF

F

v = v i ∂yi F ′ ! B′ ϕ : B → B′

r x˙ a = 0

Tp v = 0 vi : F → R

v : F → TF

◦

ϕ : F → F′

(ub , zj ; u˙ b , z˙ j ) ◦ Tϕ = (ϕb , ϕj ; ∂a ϕb x˙ a , ∂a ϕj x˙ a + ∂i ϕj y˙ i ) , F′

(ub , zj ) ϕ

Tϕ

Tϕ Vϕ : VF → VF ′

VF (ub , zj ; z˙ j ) ◦ Vϕ = (ϕb , ϕj ; ∂i ϕj y˙ i ) .

s, s˜ : B → F si ≡ y i ◦ s p:F !B

s(x) = s˜(x)

x∈B

Jx F JF ≡

p ◦ s = 11B Ts

'

Jx F .

x∈B

s:B→F Jx F

s # a i i$ x , y , ya yai ◦ js ≡ ∂a si

# a i$ x ,y F # i$ ya JF ! F F 11TB

x∈B ∂a si (x) = ∂a s˜i (x)

JF

JF ! F ! B Tp ◦ Ts = T11B = 11TB Ts : TB → TF B → T∗B ⊗F TF JF ! F dl : JF )→ T∗B ⊗TF F

α ∈ T∗B

w ∈ TF

⟨α, w⟩ ≡ ⟨α ◦ Tp, w⟩ ≡ ⟨α, Tp(w)⟩

js : B → JF x∈B JF

js

11TB JF

ν ≡ 11TF − dl : JF → T∗F ⊗ VF ⊂ T∗F ⊗ TF . F

#

xa , yi ; x˙ a , y˙ i

$

◦

# $ dl = xa , yi ; x˙ a , yai x˙ a ,

#

F

xa , yi ; x˙ a , y˙ i

$

◦

# $ ν = xa , yi ; 0, y˙ i − yai x˙ a .

dl F → T∗B ⊗F VF JF → F T∗B ⊗F VF ! F

F → JF ϕ : F → F′

ϕ : B → B′

(T ϕ)∗ ⊗ Tϕ : T∗B ⊗ TF → T∗B ′ ⊗ TF ′ . ←

F′

F

JF ⊂ T∗B ⊗F TF

ϕ

Jϕ : JF → JF ′

ϕ Jϕ

JF − −−−− → JF ′ ( ( ⏐ ⏐j(ϕ s) js⏐ ⏐ ∗ B − −−−− → B′ ϕ

# b j$ u ,z

s:B→F

←

F

′

ϕ∗ s ≡ ϕ ◦ s ◦ ϕ : B ′ → F ′

* + ← (ub , zj , zjb ) ◦ Jϕ = ϕb , ϕj , (∂a ϕj + yai ∂i ϕj ) ∂b ϕ a ◦ ϕ . B ≡ B′

←

∂b ϕ a = δab

ϕ

# $ # $ (xa , zλ , zλa ) ◦ Jϕ = xa , ϕj , ∂a ϕj + yai ∂i ϕj ≡ xa , ϕj , da ϕj .

da ϕj ≡ Ja ϕj ≡ ∂a ϕj + yai ∂i ϕj f :F →R dH f : JF → T∗B

dH f ◦ js = d(f ◦ s)

∀s : B → F .

# $ d(f ◦ s) = ∂a (f ◦ s) dxa = (∂a f ◦ s + ∂a si ∂i f ◦ s) dxa = (∂a f + yai ∂i f ) ◦ js dxa ≡ (da f ◦ js) dxa

dH f = da f dxa ≡ (∂a f + yai ∂i f ) dxa .

⇒

JF ! F ! B TJF ! TF ! TB T∗B ⊂ T∗F ⊂ T∗ JF , dH f

JF

dV f ≡ df − dH f = ∂i f (dyi − yai dxa ) : JF → T∗JF f d2H = d2V = dH dV + dV dH = 0 .

ˇ = ∂i f dyi : F → V∗F , df df

VF

JVF ∼ = VJF σ :R×B →F

∂jσ ≡ j∂σ

∂σ

(xa , yj , y˙ j , yaj , y˙ aj ) ←→ (xa , yj , yaj , y˙ j , y˙ aj ) . v : F → VF

v′ ∼ = Jv : JF → VJF v ′ = v j ∂yj + (∂λ v j + yλh ∂h v j )∂yja .

r1 : JTF → TJF

F

r

1 JTF − −−− − → TJF ( ( ⏐ ⏐Tjs JTs⏐ ⏐

JTB − −−−− → TB

#

s:B→F xa , yi , yai ; x˙ a , y˙ i , y˙ ai

$

◦ r1

# $ = xa , yi , yai ; x˙ a , y˙ i , y˙ ai − ybi x˙ ba .

F !B

k

k ∈N,

Jk F ⊂ JJk−1 F ,

Jk F ! B

s:B→F

x∈B

h, k ∈ N

jk s ≡ j jk−1s x k Jh+k F ⊂ Jh Jk F

Jk F ! Jk−1 F ! · · · ! F ! B . Jk F ! Jk−1 F DJk F = Jk−1 F × ∨k T∗B ⊗ VF ! Jk−1 F , F

F

∨k ϕ : F → F′

k ′

k Jk ϕ : J k F → Jk F JJk−1 ϕ ← Jk ϕ ◦ jk s = jk (ϕ ◦ s ◦ ϕ) ◦ ϕ s:B→F # a i$ x , y : F → Rm+n

#

A

# # b j$ u ,z

#

i $ xa , yA ,

i xa , yA

$

◦

ub , zj , zjB

ϕ : B → B′

0 ≤ | A| ≤ k ,

| A|

# $ jk s = xa , ∂A si ,

$

F′

◦

# $ Jk ϕ = ϕb , ϕj , JB ϕj ,

i JB+b ϕj ≡ ∂b JB ϕj + ∂b ϕ a · ∂iA (JB ϕj ) · yA +a , ←

0 ≤ |A|, |B | ≤ k .

f :F →B×R

F Jk f dH f : JF → T∗B

Jk F → ∨k T∗B Jk VF ∼ = VJk F

σ :R×B →F

#

k=1

∂jk σ ≡ jk ∂σ

# j j $ j j $ xa , yj , y˙ j , yA , y˙ A ←→ xa , yj , yA , y˙ j , y˙ A ,

v = v i ∂yi : F → VF v i ∂yi + JA v i ∂yiA

Jk v : Jk F → VJk F ∀k ∈ N

F

rk : Jk TF → TJk F Jk VF ∼ = VJk F r

k Jk TF − −−− − → TJk F ( ( ⏐ ⏐Tj s Jk Ts⏐ ⏐ k

Jk TM − −−−− → TM

#

i xa , y i , yA ; x˙ a , y˙ i

i i ◦ rk = y ˙A y˙ A −

$

,

s:B→F ◦ rk

# $ i = xa , y i , yA ; x˙ a , y˙ i

i ˙ aC , yB +a x

B ,C

1 ≤ |A| ≤ k ,

B

v = v a ∂xa + v i ∂yi : Jh F → TF ,

+ C = A , |C | ≥ 1 .

h∈N, k∈N

F k v(k) ≡ rk ◦ Jk v : Jh+k F → TJk F , v(k)

Jh+k F ⊂ Jk Jh F

v

# a i a i i # a i a i i i $ a i$ ◦ Jk v = x , y , v , v , yA , JA v , JA v x , y , x˙ , y˙ ; yA , x˙ aA , y˙ A , i v(k) = v a ∂xa + v i ∂yi + vA ∂yiA , i vA ≡ (v(k) )iA = JA v i −

,

B,C

1 ≤ | A| ≤ k ,

i a yB +a JC v ,

B

+ C = A , |C | ≥ 1 .

m≥1 A

#

Sa1 ...ak

$

k

1 = (A1 , . . . , Am )

a1 , . . . , a k

2 ...

m

m k = | A| ≡ m A i i=1 (0, . . . , 1, . . . , 0) ≡ (δai ) , i = 1 . . . k

a |a| = 1 A+B

= (A1 + B 1 , . . . , Am + B m ) ,

| A + B | = | A| + | B | V z

A+a

J

(t ∨ z)I +J

= (A1 , . . . , Aa + 1, . . . , Am ) t ∈ ∨h V z ∈ ∨k V |I | = h |J | = k

f :B→R

∂A f ≡ ∂A1 . . . ∂Am f , ∂Aa f ≡ ∂a . . . ∂a f (∂A f ) f |A| = 0 (∂a f ) ∂A+a f ≡ ∂a ∂A f # a i$ i x , y A , 0 ≤ | A| ≤ k yA

⎧ i y , ⎪ ⎪ ⎨ yai , ...... ⎪ ⎪ ⎩ i ya1 ...ak ,

tA+C z B+C

1

tI

k

| A| ≥ k , Aa

| A| = 1 . . .

. | A| = k

(∂a1 . . . ∂ak f ) Jk F

|A| = 0 , |A| = 1 , |A| = k .

| A| = k tA+a z B+a

M → ∧r T∗M ⊗M TM r ∈ {0} ∪ N M ζ ξ r

| A| ≤ k C

A

B

a

r s

(r+s) [[ζ, ξ]] : M → ∧r+s T∗M ⊗ TM . M

[[λ ⊗ u, µ ⊗ v]] = λ ∧ µ ⊗ [u, v] + λ ∧ (L[u]µ) ⊗ v − (L[v]λ) ∧ µ ⊗ u + (−1)r (v|λ) ∧ dµ ⊗ u + (−1)r dλ ∧ (u|µ) ⊗ v ,

λ : M → ∧r T∗M ,

µ : M → ∧s T∗M ,

[u, v]

u

u, v : M → TM ,

v

ζ = ζa1 ...arb dxa1 ∧ · · · ∧ dxar ⊗ ∂xb ,

ξ = ξa1 ...asb dxa1 ∧ · · · ∧ dxas ⊗ ∂xb ,

[[ζ, ξ]] = [[ζ, ξ]]a1 ...ar+sb dxa1∧ · · · ∧ dxar+s ⊗ ∂xb , [[ζ, ξ]]a1 ...ar+sb = ζa1 ...arc ∂c ξar+1 ...ar+sb − (−1)rs ξa1 ...asc ∂c ζas+1 ...ar+sb − r ζa1 ...ar−1 cb ∂ar ξar+1 ...ar+sc + (−1)rs s ξa1 ...as−1 cb ∂as ζas+1 ...ar+sc . M [[ζ, ξ]] = (−1)1+rs [[ξ, ζ]] , (−1)rt [[ζ, [[ξ, θ]]]] + (−1)sr [[ξ, [[θ, ζ]]]] + (−1)ts [[θ, [[ζ, ξ]]]] = 0 , θ

t F !B

r

ζ

F

ζ : F → ∧r T∗B ⊗ TF ⊂ ∧r T∗F ⊗ TF , F

r

F

∗

ζ : B → ∧ T B ⊗B TB ζ

F − −−−− → ∧r T∗B ⊗F TF ⏐ ⏐ ⏐ ⏐ 2 2

B − −−−− → ∧r T∗B ⊗B TB ζ

ζab1 ...ar ≡ ζ ba

1 ...ar

:B→R

* + [[ζ, ξ]] = dxa1∧ · · · ∧ dxar+s ⊗ [[ζ, ξ]]a1 ...ar+sb ∂xb + [[ζ, ξ]]a1 ...ar+si ∂yi , [[ζ, ξ]]a1 ...ar+sb = [[ζ, ξ]]a1 ...ar+sb

,

[[ζ, ξ]]a1 ...ar+si = ζac1 ...ar ∂c ξar+1 ...ar+si − (−1)rs ξac 1 ...as ∂c ζas+1 ...ar+si + ζaj 1 ...ar ∂j ξar+1 ...ar+si − (−1)rs ξaj 1 ...as ∂j ζas+1 ...ar+si − r ζai 1 ...ar−1 c ∂ar ξar+1 ...ar+sc + (−1)rs s ξai 1 ...as−1 c ∂as ζas+1 ...ar+sc . 0

ζ

F → ∧r T∗B ⊗F VF

ξ

[[ζ, ξ]]

# [[ζ, ξ]] = ζa1 ...arc ∂c ξar+1 ...ar+si + ζa1 ...arj ∂j ξar+1 ...ar+si

$ − (−1)rs ξa1 ...asj ∂j ζas+1 ...ar+si + (−1)rs s ξa1 ...as−1 ci ∂as ζas+1 ...ar+sc · · dxa1∧ · · · ∧ dxar+s ⊗ ∂yi .

F !B κ : F → JF

dl dl ◦ κ : F → T∗B ⊗ TF ⊂ T∗F ⊗ TF . F

F

κ ≡ κ ◦ dl κ = dxa ⊗ (∂xa + κai ∂yi ) , κ

κai : F → R . ν : JF → T∗F ⊗F VF

νκ ≡ ν ◦ κ : F → T∗F ⊗ VF ⊂ T∗F ⊗ TF , F

F

νκ = (dyi − κai dxa ) ⊗ ∂yi . κ

T∗B ⊂ T∗F F

νκ

VF ⊂ TF

TF ! F

F 0 → VF → TF → TB × F → 0 . M

u : B → TB

κ

u⌋κ : F → TF , u⌋κ = ua (∂xa + κai ∂yi ) v : F → TF v ⌋νκ : F → VF , v ⌋νκ = (v i − v a κai ) ∂yi F

TB JF ! F

κab = δab

κ − κ′ : F → T∗B ⊗F VF B B r

∗

ξ : F → ∧ T B ⊗F VF ξ

r r+1

d[κ]ξ ≡ [[κ, ξ]] : F → ∧r+1 T∗B ⊗VF . F

d[κ]ξ = (d[κ]ξ)a1 ...ar+1i dxa1∧ · · · ∧ dxar+1 ⊗ ∂yi ,

i

(d[κ]ξ)a1 ...ar+1i = ∂a1 ξa2 ...ar+1i + κa1j ∂j ξa2 ...ar+1i − (−1)r ξa1 ...arj ∂j κar+1 = ∂a1 ξa2 ...ar+1i + κa1j ∂j ξa2 ...ar+1i − ∂j κa1i ξa2 ...ar+1j .

r=1

⇒

r=0

⇒

# $ [[κ, ξ]] = ∂a ξbi + κaj ∂j ξbi − ∂j κai ξbj dxa ∧ dxb ⊗ ∂yi , # $ [[κ, ξ]] = ∂a ξ i + κaj ∂j ξ i − ∂j κai ξ j dxa ⊗ ∂yi .

s:B→F ∇[κ]s ≡ js − κ ◦ s = Ts⌋νκ : B → T∗B ⊗VF , F

∇s = ∇a si dxa ⊗ ∂yi

∇s ∇a si ≡ (∇s)ai = ∂a si − κai ◦ s . u : B → TB ∇u s ≡ u⌋∇s : B → VF , ∇u s = ua ∇a si ∂yi κ

ρ ≡ −d[κ]κ ≡ −[[κ, κ]] : F → ∧2 (T∗B) ⊗VF . F

u, v : B → TB

ρ(u, v) = [u, v]⌋κ − [u⌋κ , v ⌋κ] , u (→ u⌋κ

ρ[κ] = ρabi dxa ∧ dxb ⊗ ∂yi = ρabi dxa ⊗ dxb ⊗ ∂yi , ρabi = −∂a κbi + ∂b κai − κaj ∂j κbi + κbj ∂j κai .

∂κ

κ∂ κ

d[κ]ρ = −[[κ, [[κ, κ]]]] = 0 , c:R→B

p◦C =c, C

F !B

κ ←

y ∈ Fc(t0 ) ≡ p (c(t0 )) t0 ∇dc [κ]C = 0 .

c

κ

y

c C

C:R→F

F !B

˜ ∇dc C

˜:B→F C

E!B EndE ≡ E ⊗B E ! B

E∗ ! B E 11E : B → EndE ≡ E ⊗ E ∗

∗

VE ∼ = E ×E B

B

V (VE)x = T(Ex ) = Ex ×Ex ,

∇s : B → T∗B ⊗B E

s

V ×V

∀x ∈ B .

s:B→E

κ : E → JE (·) : R × E → E

∇0 = 0 ,

j0 : B → JE ,

B

0:B→E

∇(s + s′ ) = ∇s + ∇s′ ,

(+) : E ×B E → E

∇(f s) = df ⊗ s + f ∇s .

J(·) : R×JE → JE ,

J(+) : JE × JE → JE , B

JE ! B JE ! E κ : E → JE LCE ! B LCE ⊂ JE ⊗E ∗ B

11E : B → EndE ≡ E ⊗B E ∗ κ : B → LCE E → JE LCE ! M DLCE ≡ T∗B ⊗ EndE ! B . B

κ0 EndE ! B

B → T∗B ⊗B EndE

L ⊂ EndE B → T∗B ⊗B L

# i$ y

E∗ ! B

κ : E → JE κai = κaji yj ,

κaji : B → R .

∂j κai = κaji ξ : E → ∧r T∗B ⊗E VE

r

r=1

⇒

r=0

⇒

$ # [[κ, ξ]] = ∂a ξbi + κajh yh ∂j ξbi − κaji ξbj dxa ∧ dxb ⊗ ∂yi , # $ [[κ, ξ]] = ∂a ξ i + κajh yh ∂j ξ i − κaji ξ j dxa ⊗ ∂yi . κ

# $ ρabi = ρabij yj = −∂a κbij + ∂b κaij + κaih κbhj − κbih κahj yj . ρ

ρ : B → ∧2 T∗B ⊗ EndE ≡ ∧2 T∗B ⊗ E ⊗ E ∗ , B

B

B

ρ = ρabij dxa ∧ dxb ⊗ yi ⊗ yj .

ρ(u, v)s = ∇u ∇v s − ∇v ∇u s − ∇[u,v] s u, v : B → T∗B c:R→B

E!B

s:B→E

(κajk )

κa : B → EndF ∇a s = ∂a s − κa s

ρab = −∂a κb + ∂b κa + κa κb − κb κa ≡ κ[a,b] + [κa , κb ]

E!B

E′ ! B ⊗ : E × E ′ → E ⊗ E ′ : (y, y ′ ) (→ y ⊗ y ′ B

B

J⊗ : JF ×B JF ′ → J(F ⊗B F ′ ) J⊗ ◦ (js, js′ ) = j(s ⊗ s′ )

κ

κ

′

s′ : B → E ′ E

s:B→E

E′

κ ⊗ κ′ : E ⊗ E ′ → J(E ⊗ E ′ ) B

B

J⊗ ◦ (κ, κ′ ) = (κ ⊗ κ′ ) ◦ ⊗ .

κ

κ′ κ ⊗ κ′

∇u (s ⊗ s′ ) = (∇u s) ⊗ s′ + s ⊗ ∇u s′ , ∀ u : B → TB ,

s:B →E,

′

′

s′ : B → E ′ .

′

(κ ⊗ κ′ )aii jj ′ = κaji δi j ′ + δij κ′ ai j ′ . E !B ∗

E!B

∗

∇[κ ⊗ κ ]11E = 0 .

∗

κ 11E : B → E ⊗ E ∗

∗

κ ∗

∇u ⟨σ, s⟩ ≡ u.⟨σ, s⟩ = ⟨∇u σ, s⟩ + ⟨σ, ∇u s⟩ , u : B → TB s : B → E

κ

σ : B → E∗ ∗ κ

∗

κ aij = −κaji .

∗

ρ abji = −ρabij . κ E!B ⊗B (E ⊕B E ∗ ) ! B

κ⊗ κ⊗ ∧n E ∗ ! B

ˆ κ E!B

n

∗

(ˆ κ)a = κ aii = −κaii .

M Γ # a$ x

m TM ! M Γ = dxa ⊗ (∂xa + Γabc x˙ c ∂ x˙ b ) ,

#

xa , x˙ a

$

Γabc : M → R .

Γ

R = Rabcd dxa ∧ dxb ⊗ ∂xc ⊗ dxd : M → ∧2 T∗M ⊗ TM ⊗ T∗M , M

Ric = Ricad dxa ⊗ dxd : M → T∗M ⊗ T∗M , M

M

Ricad ≡ Rad = Rabbd .

Γ

T : M → TM ⊗ ∧2 T∗M , M

T (u, v) = ∇u v − ∇v u − [u, v] ,

u, v : M → TM ,

T cab = −Γacb + Γbca

⇒

Γbca = Γacb + T cab .

M → T∗M ⊗ TM ⊗ T∗M ≡ T∗M ⊗ End(TM ) , TM ! M Γ = (Γ +

(Γ +

1 2

T )acb = Γacb + =

1 2

1 2

1 2

T) −

1 2

T,

T cab = Γacb +

1 2

(Γbca − Γacb )

(Γacb + Γbca ) .

Γ′ ≡ Γ +

1 2

T′ = 0

T

Ric′ba = Ric′ab M g = gab dxa ⊗ dxb : M → T∗M ⊗ T∗M . M

TM ! M

Γ

∇[Γ]g = 0

Γ

3 4 Γacb = − 12 gce (∂a gbe + ∂b gea − ∂h gab ) + T cab − T bac − T abc ,

Tabc ≡ T ebd gea gdc .

g

η=

√

|g|

dm x : M → ∧m T∗M ,

|g| = | det g| ,

g# (η, η) = sgn det g , ∇η = 0 ⟨R⟩ ≡ gad Rad ≡ Rabba : M → R .

g T =0

(M , g)

.

R+ × L → L

.

. . .

L

.

.

R+

L

R+ A

A

R+

R+ ∪{0}

R

Gauge Field Theory in Natural Geometric Language: A revisitation of mathematical notions of quantum physics. Daniel Canarutto, Oxford University Press (2020). © Daniel Canarutto. DOI: 10.1093/oso/9780198861492.001.0001

R+ V

V+ ⊂ V + V ⊗V = V⊗V U

a∈R

V+ ⊗ R = V

m∈N

U +

V

u∈U

⊗m (U1/m ) ∼ =U f : U → R+

U1/m

p 1/m Up/m ≡ ⊗p (U1/m ) ∼ , = (⊗ U)

U

−1

∗

≡U ,

U

1/m

U ⊗ U∗ ∼ = R+ m

f (au) = a−1/m f (u)

p∈N,

Ur

r∈Q

u∈U u−1 ≡ u∗ ∈ U−1

u1/m ∈ U1/m (u1/m )m = u uv ≡ u ⊗ v

v

T

L

S ≡ S[d1 , d2 , d3 ] ≡ Td1 ⊗ Ld2 ⊗ Md3 , s∈S

M di ∈ Q .

E!B

S⊗E ! B

E E′ ! B D : Sec(B, E) → Sec(B, E ′ )

D : Sec(B, S ⊗ E) → Sec(B, S ⊗ F ) : σ (→ Dσ ≡ s ⊗ D(s−1 |σ) , λ : B → S ⊗ ∧p T∗B

s∈S p p∈N dλ : B → S ⊗ ∧p+1 T∗B v L[v]λ : B → S ⊗ ∧p T∗B

B B Γ

E!B σ : B → S⊗E # $ Γ yi E # $ s ⊗ yi s∈S

S⊗E ! B ∇σ : B → S ⊗ T∗B ⊗ E σ:B→E

σ′ : B → S ⊗ E s:B→S

c ∈ T−1 ⊗ L

•

! ∈ T−1 ⊗ L2 ⊗ M

• •

G

∈ T−2 ⊗ L3 ⊗ M−1

e ∈ T−1 ⊗ L3/2 ⊗ M1/2

•

m∈M

•

L2 T∼ =L

M∼ = L−1

L

f :A→C

A

f¯ : A → C : a (→ f¯(a) ≡ f (a) , a ∈ A . V

W

V ⊗W

Lin(V , W ) Lin(V , W ) V →W

V V ⋆ ≡ Lin(V , C) ,

V ⋆ ≡ Lin(V , C) ,

V →C

V

V ≡ V ⋆ ⋆ ≡ (V ⋆ )⋆ .

f :V →W

f (iv) = −if (v)

V V

c

v

c¯ v

v∈V

¯, V⋆↔V⋆:λ↔λ ¯=λ λ

¯∈V⋆ λ V

λ∈V⋆ V

V ↔ V : v ↔ v¯ . ⋆⋆ ∼ ⋆ ⋆ V ∼ , =V =V ⋆⋆ V ≡ V ⋆⋆ ∼ . =V

V⋆

V ⋆ → C : λ (→ ⟨λ, v⟩ v¯ ∈ V

v∈V V ⋆ → C : α (→ ⟨α, ¯ v⟩ → C : α (→ ⟨α, v¯⟩ ≡ α(v) V ↔V ,

V⋆↔V⋆,

⋆

V

⋆

⋆ ⋆⋆ V⋆∼ ≡ (V ⋆ ⋆ )⋆ ≡ V ⋆ . =V

⋆ Lin(V , W ) ∼ = W ⊗V ,

#

$ bα 1 ≤ α ≤ dim V

¯α˙ ≡ bα , b

v = v α bα , α

λ = λα b ,

#

bα

¯α˙ ≡ bα , b

¯α˙ , v¯ = v¯α˙ b ¯α˙

¯=λ ¯ α˙ b , λ

⋆ Lin(V , W ) ∼ = Lin(V , W ) ∼ = W ⊗V .

$

V V

#

bα

$

#

v¯α˙ ≡ v α ,

¯ α˙ ≡ λα , λ v α˙

bα

$

V⋆ V⋆∼ =V⋆

v ∈ V , v¯ ∈ V ,

¯∈V⋆. λ∈V⋆, λ v¯α˙

λα˙

¯ α˙ λ

T γ˙ ϕ˙ Tαβ δ˙ϵ

¯α˙β˙γ δϵ˙ ϕ T

¯ T f ∈ Lin(V , W ) ∼ = W ⊗V

f¯ ∈ Lin(V , W )

⋆

⋆ f¯ ∈ Lin(V , W ) ∼ = W ⊗V ∼ = W ⊗V ⋆ .

V⋆ V

V V∗

∗

R

V →R

V V⋆⊂V∗

† : V ⊗ V → V ⊗ V : w (→ w†

(u ⊗ v¯)† = v ⊗ u ¯ † ◦ † = 11

V ⊗V

±1

¯ V ) ⊕ i (V ∨ ¯V). V ⊗ V = (V ∨ w† = ±w

w

w = 12 (w + w† ) + 12 (w − w† ) . w ℑw ≡

¯B˙ ∈ V ⊗ V , w = wAB˙ bA ⊗ b

1 2i

(w − w) ¯

ℜw ≡

1 2

(w + w) ¯

(wAB˙ )

w w ¯

B˙A

= ±w

AB˙

V ⋆ ⊗V ⋆

⋆ ¯ V )∗ ∼ ¯V⋆ , (V ∨ =V ∨

⋆ ¯ V )∗ ∼ ¯V⋆, (i V ∨ = iV ∨

∗

¯V⋆ h∈V⋆∨

h♭ : V → V ⋆ : v¯ (→ v¯♭ ≡ h(¯ v, ) , ⟨¯ v ♭ , u⟩ ≡ h(¯ v , u)

v (→ h(¯ v , v)

¯A˙ ⊗ bA h = hA˙A b ¯A˙ ⊗ bA ∈ V ∨ ¯V , h−1 = hA˙A b ˙ hC ˙A hC ˙B = δAB hA˙C hB˙C = δAB ˙

¯ (→ λ ¯ # ≡ h−1 (λ, ¯ ), h# : V ⋆ → V : λ ¯ # ⟩ ≡ h−1 (λ, ¯ µ) ⟨µ, λ

h♭

h#

¯ # : V ⋆ → V : λ (→ λ# ≡ h ¯ −1 (λ, ) = h−1 ( , λ) , h ¯ ♭ : V → V ⋆ : v (→ v ♭ ≡ h ¯ (v, ) = h( , v) . h

v¯A ≡ (¯ v ♭ )A = hA˙A v¯A˙ ,

¯ A ≡ (λ ¯ # )A = hA˙A λ ¯ A˙ , λ

h# −1

h

X ∈ EndV ∼ = V ⊗V ⋆ ⋆ X ∗ ∈ EndV ⋆ ∼ = V ⊗V ,

⋆ ¯ ∈ EndV ∼ X = V ⊗V ,

⋆ ¯ ∗ ∈ EndV ⋆ ∼ X = V ⊗V .

f

h ¯ ♭ ∈ EndV ¯∗ ◦ h X † ≡ h# ◦ X

˙ ¯ DC (X † )AB = hC ˙A hD˙B X ˙.

X† (X ) = X

h(X † u, v) = h(u, Xv) ∀ u, v ∈ V

† †

X ∈ EndV

X † = −X

h

X ∈ EndV ⟨(Xu)† , Xv⟩ = ⟨u† , v⟩ ∀ u, v ∈ V

h

X † ◦ X = f ◦ X † = 11V |det X|2 = 1

n

V

h

n2 ⋆ EndV ≡ Lin(V , V ) ∼ = V ⊗V ,

G

: EndV × EndV → C : (X, Y ) (→ Tr(X ◦ Y ) = ⟨X ∗ , Y ⟩ ,

⋆ ∗ EndV → EndV ⋆ ∼ = (EndV ) : X (→ X .

2n2

X† = X

EndV

ℜG : EndV × EndV → R : (X, Y ) (→ ℜ Tr(X ◦ Y ) .

h L ⊂ EndV h iL ⊂ EndV

n

L

EndV = L ⊕ iL , 1 2

X ∈ EndV

iL

L

G

(X − X † ) +

1 2

(X + X † ) (n2 , n2 )

ℜG h#

V

⋆

EndV ⋆ = L′ ⊕ iL′ . (X ∗ )† = (X † )∗

X∈L

X ∈ iL

(X ∗ )† = (X † )∗ = ∓X ∗ .

G

♭

: EndV → EndV ⋆ : X (→ X ∗

iL

L

G

♭

|L : L (→ L′ ,

G

♭

|iL : iL (→ iL′ . ˜ h

h

EndV

˜ : EndV × EndV → C : (X, ¯ Y ) (→ Tr(X † ◦ Y ) h ˜ ♭ (X) = (X † )∗ h

#

bα

$

# $ lI

V

lI = lIαβ bα ⊗ bβ ,

∓G

iL

L

# I$ l

L

lI = lIαβ bα ⊗ bβ .

cIJH ≡ ⟨lI , [lJ , lH ]⟩ = lIαβ (lJαγ lHγβ − lHαγ lJγβ ) ,

cIJH + cIHJ = 0 , !

bα

"

! " lI

cKIL cLJH + cKJL cLHI + cKHL cLIJ = 0 . V !M

lIαβ

cIJH

L∗

Z = X + i Y ∈ L ⊕ i L = EndV

Z = Z αβ bα ⊗ bβ = Z I lI ≡ (X I + i Y I ) lI = (X I + i Y I ) lIαβ bα ⊗ bβ , Z αβ = Z I lIαβ , XI, Y I ∈ R

Z I = ⟨lI , Z⟩ = lIαβ Z αβ , Z, Z ′ ∈ EndV L∗

L

[Z, Z ′ ] = cIJH Z J Z ′ H lI G

L

G

cIJH ≡ GIL cLJH = G ♭ (lI , [lJ , lH ]) .

G (X, [Y, Z])

= G ([X, Y ], Z) .

l†I = −lI G

∗ JH I

c

J

♭

(lI ) ≡ l∗I = GIJ lJ = lIαβ bβ ⊗ bα ∈ L∗

H

≡ ⟨lI , [l , l ]⟩

lI = lIαβ bα ⊗ bβ ∈ L∗ ∗

cIJH = −cIJH .

V n

g : V ×V → R : (u, v) (→ g(u, v) . ˜ h

L V

−G

L

[lJ , lH ]∗ = cKJH l∗K = cKJH GKI lI ≡ cIJH lI ∗ ∗ [lJ , lH ]∗ = [l∗H , l∗J ] = − = [l∗J , l∗H ] = −[GJL lL , GHK lK ] = −GJL GHK cILK lI = − cIJH lI

2n

V ∧V ≡

n !

r=0

∧r V ≡ R ⊕ V ⊕ ∧2 V ⊕ · · · ⊕ ∧n V , (a, b) (→ a ∧ b ∧V × ∧V → ∧V ,

uv = u ∧ v + g(u, v) ,

u, v ∈ V .

ClV ≡ Cl(V , g)

∧V

Z2

ClV

uv + vu = 2g(u, v)

vv = g(v, v) V

uv ∧V [i

v1 ∧ v2 ∧ · · · ∧ v r =

i

i ]

ϵi1 i2 ...ir ≡ δ1 1 δ22 · · · δrr

v1 , v 2 , . . . , v r ∈ V 1 r!

,

i1 i2 ...ir

ϵi1 i2 ...ir vi1 vi2 · · · vir .

u, v, w, z ∈ V uvw = u ∧ v ∧ w + g(v, w) u − g(u, w) v + g(u, v) w , uvwz = u ∧ v ∧ w ∧ z

+ g(w, z) u ∧ v − g(v, z) u ∧ w + g(v, w) u ∧ z + g(u, z) v ∧ w − g(u, w) v ∧ z + g(u, v) w ∧ z

+ g(u, z) g(v, w) − g(u, w) g(v, z) + g(u, v) g(w, z) , (u ∧ v) (w ∧ z) = u ∧ v ∧ w ∧ z − g(v, z) u ∧ w + g(v, w) u ∧ z

+ g(u, z) v ∧ w − g(u, w) v ∧ z + g(u, z) g(v, w) − g(u, w) g(v, z) .

a ∈ ∧r V V

b ∈ ∧s V ab r+s, r+s−2 , . . . , |r−s| g

η ∈ ∧n V ∗

∗

η# ≡ g# (η) = ± η ∈ ∧n V η

g

±∗v = v η# = ±η# v , # #

η η ∗ : ∧r V → ∧n−r V

v∈V ,

= ±1 , V

∗v ∈ ∧n−1 V

g A

γ:V →A γ(v) γ(v) = g(v, v) 11 ,

v∈V ,

ˆ : ClV → A γ

γ S

. . .

.

.

.

.

S (∧2 S)⋆ ⟨λ ∧ λ′ , s ∧ s′ ⟩ ≡

1 4

# $ ⟨λ, s⟩⟨λ′ , s′ ⟩ − ⟨λ, s′ ⟩⟨λ′ , s⟩ ,

⟨λ ∧ λ′ , s ∧ s′ ⟩ =

1 2

(λ ∧ λ′ )⌋(s ∧ s′ ) =

1 4

∧2 S ∧2 S ⋆ s, s′ ∈ S , λ, λ′ ∈ S ⋆ .

(λ ∧ λ′ )|(s ∧ s′ ) . w−1 ≡ w ∈ ∧2 S ⋆ ∗

w ∈ ∧2 S\{0} ∗ ⟨w, w⟩ = 1

w# : S ⋆ → S : λ (→ w# (λ) ≡ w(λ, ) : λ′ (→ w(λ, λ′ ) , w♭ : S → S ⋆ : s (→ w♭ (s) ≡ w(s, ) : s′ (→ w(s, s′ ) , ∗

∗

∗

w# = −(w♭ )−1 .

Gauge Field Theory in Natural Geometric Language: A revisitation of mathematical notions of quantum physics. Daniel Canarutto, Oxford University Press (2020). © Daniel Canarutto. DOI: 10.1093/oso/9780198861492.001.0001

¯ (∧2 S) ⊂ ∧2 S ⊗ ∧2 S . (∧2 S) ∨ w⊗w ¯

¯ (∧2 S) (∧2 S) ∨

w ∈ ∧2 S

3 4 ¯ (∧2 S) + ⊂ (∧2 S) ∨ ¯ (∧2 S) , L2 ≡ (∧2 S) ∨

L ≡ (L2 )1/2 ,

U ≡ L−1/2 ⊗ S . U U

Lr

r

Q ≡ ∧2 U = L−1 ⊗ ∧2 S ,

−2 2 2 ⋆ ¯ (∧2 S) ∼ ¯ (∧2 S ⋆ )] ⊗ [(∧2 S) ∨ ¯ (∧2 S)] Q⋆ ⊗ Q⋆ ∼ = L ⊗ (∧ S) ∨ = [(∧ S ) ∨ 2 ∼ ¯ (∧2 S)]⋆ ⊗ [(∧2 S) ∨ ¯ (∧2 S)] . = [(∧ S) ∨

Q

Q

Q⋆ ⊗ Q⋆

⋆

U

# $ bA ⊂ S ,

# A$ b ⊂ S⋆ ,

A

= 1, 2 .

w ≡ ϵAB bA ∧ bB = 2 b1 ∧ b2

∧2 S ,

∗

∧2 S ⋆ ,

w ≡ ϵAB bA ∧ bB = 2 b1 ∧ b2 ll ≡

√

¯ w⊗w

L,

ϵAB ≡ δ1A δ2B − δ1B δ2A

B B A ϵAB ≡ δA 1 δ2 − δ1 δ2

#

$ # $ zA ≡ ll−1/2 ⊗ bA ⊂ U ,

(zA ) ≡ (ll1/2 ⊗ bA ) ⊂ U ⋆ .

ϵ ≡ ll ⊗ w = ϵAB zA ∧ zB ∈ Q⋆ ≡ ∧2 U ⋆ , ∗

∗

ϵ ≡ ll−1 ⊗ w = ϵAB zA ∧ zB ∈ Q ≡ ∧2 U , ∗

ϵ

ϵ

ϵ♭ : U → U ⋆ : u (→ u♭ ≡ ϵ♭ (u) ≡ ϵ(u, ) : u′ (→ ϵ(u, u′ ) , ϵ# : U ⋆ → U : λ (→ λ# ≡ ϵ# (λ) ≡ ϵ−1 (λ, ) : λ′ (→ ϵ−1 (λ, λ′ ) , ϵ# = −(ϵ♭ )−1

#

b′A

ϵ

$

ϵ′ = eit ϵ t ∈ R

S

u♭ = uA zA ≡ ϵAB uB zA ,

u = uB z B ∈ U ,

λ# = λA zA ≡ ϵAB λB zA ,

ϵAB ϵ

BC

C A

= −δ

⇒

5

ϵ ∈ Q⋆ U ⊗U

λ = λB z B ∈ U ⋆ ,

uA ϵAB ϵBC = −uC ,

λA ϵAB ϵBC = −λC .

¯ ∈ Q⋆ ⊗ Q⋆ g ≡ ϵ⊗ϵ

¯(¯ g(u ⊗ v¯, u′ ⊗ v¯′ ) = ϵ(u, v) ϵ u, v¯′ ) .

ϵ g

g U ⊗ U ↔ U⋆ ⊗ U⋆

#

zA

¯A˙B˙ zA ⊗ ¯zA˙ ⊗ zB ⊗ ¯zB˙ , g = gAA˙BB˙ zA ⊗ ¯zA˙ ⊗ zB ⊗ ¯zB˙ ≡ ϵAB ϵ w = wAA˙ zA ⊗ ¯zA˙ ∈ U ⊗ U

# $ ¯A˙B˙ wAA˙ wBB˙ = 2 det wAA˙ , g(w, w) = ϵAB ϵ

w♭ ≡ g(w, ) = gAA˙BB˙ wAA˙ zB ⊗ ¯zB˙ ∈ U ⋆ ⊗ U ⋆ .

$

# $ bA

¯ U ⊂ U ⊗U , H≡U∨ H

g (+ − − −) # $ τλ ⊂ H

τλ ≡ τλAA˙ zA ⊗ ¯zA˙ ≡ σλAA˙ ϵ #

√1 2

σλAA˙ zA ⊗ ¯zA˙ ,

λ = 0, 1, 2, 3 ,

λ g(τλ , τµ ) = 2 δ0λ δ0µ − δλµ g

$ ⋆ ¯ U⋆ ⊂ U⋆ ⊗ U⋆ , τλ ⊂ H ∗ ∼ =U ∨

τλ ≡ τλAA˙ zA ⊗ ¯zA˙ ≡ ¯A˙B˙ σµBB˙ , σλAA˙ ≡ gλµ ϵAB ϵ

λ A √1 σ AA˙ z 2

⊗ ¯zA˙ ,

¯A˙B˙ . gλµ = τλAA˙ τµBB˙ ϵAB ϵ ∗

∗

H∗

¯ g# = ϵ ⊗ ϵ ¯A˙B˙ . gλµ = τλAA˙ τµBB˙ ϵAB ϵ

w ∈ U ⊗U

N ⊂H

g(w, w) = 0 w = u ⊗ v¯

U ⊗U

u, v ∈ U ±u ⊗ u ¯

H

H N + ≡ {u ⊗ u ¯, u ∈ U } ,

N − ≡ {−u ⊗ u ¯, u ∈ U } .

H

2×2

#

$ #! 1 σ0 , σ1 , σ2 , σ3 = 0

0" 1 ,

!0 1

1" 0 ,

!0 i

−i " 0 ,

!1

0" 0 −1

$

.

∗

⋆

#

zA

# $ eλ u, v ∈ U g(τ, τ ) = τ0 ≡ √12 (z1 ⊗ ¯z1 + z2 ⊗ ¯z2 )

√

# $ eλ ⊂ H

$

e #0 $ τλ

τ = u⊗u ¯ + v ⊗ v¯

2 |ϵ(u, v)|

τ τ ♭ ∈ H∗

U U

C

H

N

R

S

U U → N : κ (→ κ ⊗ κ ¯

R→C

w ∈ U ⊗U

w♭ ≡ g♭ (w) ∈ U ⋆ ⊗ U ⋆

¯ (→ w⌋λ ¯, U⋆ → U : λ

U → U ⋆ : u (→ u⌋w♭ .

γ : U ⊗ U → End(U ⊕ U ⋆ ) : w (→ γ[w] ¯ ≡ γ[w](u, λ)

√ # $ ¯ , u ⌋w ♭ . 2 w ⌋λ

w = p ⊗ q¯ √ # $ √ # $ ¯ = 2 ⟨λ, ¯ q¯⟩ p , ⟨p♭ , u⟩ q¯♭ ≡ 2 ⟨λ, ¯ q¯⟩ p , ϵ(p, u) ϵ ¯♭ (¯ γ[p ⊗ q¯](u, λ) q) .

γ

γ[w] ◦ γ[(w′ ] + γ[w′ ] ◦ γ[w] = 2 g(w, w′ ) 11 w, w′

(τ0 +τ1 , τ0 −τ1 , τ0 +τ2 , τ0 +τ3 ) U ⊗U

U ⊗U

(τλ )

H

U ⊗U

U

ϵ(p, q) r♭ + ϵ(q, r) p♭ + ϵ(r, p) q ♭ = 0 p, q, r ∈ U

#

$# $ # $ ¯ = 2 ϵ(p, r) ϵ ¯ ¯(¯ γ[p ⊗ q¯] ◦ γ[r ⊗ s¯] + γ[r ⊗ s¯] ◦ γ[p ⊗ q¯] u, λ q , s¯) u, λ

# $ ¯ . = 2 g(p ⊗ q¯, r ⊗ s¯) u, λ

H

γ

W ≡ U ⊕ U⋆

γ : ∧H → EndW γ[w ∧ y] =

1 2

γ : H → EndW

(γ[w] ◦ γ[y] − γ[y] ◦ γ[w]) .

γ(∧H) ⊂ EndW η ∈ ∧4 H ∗ ∗ η ∈ ∧4 H ∗ γη ≡ γ[ η ] ∈ EndW

g

γ[w] ◦ γη = −γη ◦ γ[w] , ¯ = i (u , −λ) ¯ , γη (u, λ) U

U⋆

w∈H, ¯ ∈ W. ψ = (u, λ) ±i

γη u = 12 (ψ − iγη ψ) ,

¯ = 1 (ψ + iγη ψ) . λ 2

W ≡ U ⊕ U⋆ ¯ ↔ (λ, ¯ u) . ex : U ⊕ U ⋆ ↔ U ⋆ ⊕ U : (u, λ) ex

W ←→ W ⋆ ¯ ψ ′ ) (→ k(ψ, ¯ ψ ′ ) ≡ ⟨ex(ψ), ¯ ψ′ ⟩ . k : W × W → C : (ψ, ∗

η = −η# ≡ −g# (η)

"

¯ ψ ′ ≡ (u′ , λ ¯′) ψ ≡ (u, λ) ¯ ψ ′ ) = ⟨λ, u′ ⟩ + ⟨λ ¯′, u k(ψ, ¯⟩ .

k ¯ ) k♭ : W → W ⋆ : ψ¯ (→ k(ψ, W → W⋆

¯ ∈W ψ ≡ (u, λ)

¯ = (λ, u ψ¯♭ ≡ k♭ (ψ) ¯) ∈ W ⋆ . ¯ γ[w]ψ ′ ) , k(γ[w]ψ, ψ ′ ) = k(ψ,

2

γ[w] ∈ EndW

ϵ∈∧U W

∗

⋆

w∈H,

k

¯) ∈ ∧ W ⋆ ( ϵ, ϵ 2

ψ¯ ¯ ∈ W⋆ ex(ψ) ψ

ψ¯ ¯ ψ¯♭ ≡ k♭ (ψ) ψ¯

ψ (→ ψ †

W

γ k W U

U⋆

#

$ zA ⊂ U √ # $ ¯ = 2 wAA˙ λ ¯ A˙ zA , uA wAA˙ ¯zA˙ , γ[w](u, λ)

¯A˙B˙ ϵAB wBB˙ . wAA˙ ≡ ϵ

γ

γ[w] ◦ γ[w] = g(w, w)11W ,

w ∈ U ⊗U ,

¯C ˙A˙ ϵCA = (w11 w22 − w12 w21 ) δB wBA˙ wAA˙ = wBA˙ wCC ˙ ϵ A = wAB˙ wAA˙ =

1 2

1 2

g(w, w) δB A ,

˙ g(w, w) δA B˙ ,

# $ ¯ = 2 wBA˙ wAA˙ uA zB , wAB˙ wAA˙ λ ¯ A˙ ¯zB˙ = g(w, w) (u, λ) ¯ . γ[w] ◦ γ[w](u, λ) #

$ # $ ζα ≡ (ζ1 , ζ2 , ζ3 , ζ4 ) ≡ z1 , z2 , −¯z1 , −¯z2 ⊂ W ≡ U ⊕ U ⋆ ,

ζ′1 ≡

1 √ (z 2 1

, ¯z1 ) =

ζ′3 ≡

1 √ (z 2 1

, −¯z1 ) =

√1 (ζ1 2

− ζ3 ) ,

1 √ ( ζ1 2

+ ζ3 ) ,

#

#

τλ

z1 ≡ (z1 , 0)

# ′$ ζα ⊂ W α = 1, 2, 3, 4

ζ′2 ≡

√1 (z2 2

, ¯z2 ) =

ζ′4 ≡

√1 (z2 2

, −¯z2 ) =

$ # $ γλ ≡ γλαβ

γλ ≡ γ[τλ ] ⊂ EndW ,

$

√1 (ζ2 2

− ζ4 ) ,

√1 (ζ2 2

+ ζ4 ) .

λ = 0, 1, 2, 3 ,

γη = γ0 ◦ γ1 ◦ γ2 ◦ γ3 , iγ5 ≡ γ0 γ1 γ2 γ3

k

(+ + − −)

γ0

W

γ0 τ0

2

ω∈∧U

∧2 U

⋆

ω = eit ϵ ,

t ∈ (0, 2π) ,

±1

ϵ ∈ ∧2 U ⋆

¯ ω # = e−it ϵ# ω ¯ = e−it ϵ ω

# $ # $ ¯ (→ Cω ψ ≡ ω # (λ), −¯ Cω : W → W : ψ ≡ (u, λ) ω ♭ (¯ u) = e−it λ# , −¯ u♭ , Cω ◦ Cω = 11W ,

γ[w] ◦ Cω + Cω ◦ γ[w] = 0 ,

w∈H.

ψ (→ ψ c ≡ ηc C ψ¯T

ψ¯T

ηc ψ¯

τ0 ¯ = eit Cω (γ0 (ψ)) C(ψ)

C:W →W γ0 ≡ γ[τ0 ] Cω ◦ Cω = 11W Cω ψ= u = e−it λ#

1 2

(ψ + Cω ψ) +

$

(ψ − Cω ψ) . Cω (ψ) = ψ

¯ U⋆ h ∈ U⋆ ∨

¯⊗λ ± µ h = ±λ ¯⊗µ , λ, µ

1 2

¯ ∈W ψ ≡ (u, λ) ¯ = −e−it u λ ¯♭

U

#

W ¯ ∈W ψ ≡ (u, λ)

±1

h

λ, µ ⊂ U ⋆ ,

h

U⋆

h g

H

¯ ∈ H∗ ∼ ¯ U⋆ h = U⋆ ∨

¯ # ± µ# ⊗ µ ¯ # ≡ g# (h ¯ ) = ±λ# ⊗ λ h ¯# ∈ H

h

λ# ≡ ϵ# (λ) ∈ U h

¯# h

¯# h

µ∝λ

¯#

h

¯#, h ¯ # ) = g# (h ¯, h ¯ ) = ±2 | ϵ (λ, µ)|2 , g(h ∗

(+ +)

h

U h# ≡ g# (h) h ¯#, h ¯ # ) = g# (h ¯, h ¯) = 2 g(h

g h−1

(− −)

U⋆ τ ≡

1 √ 2

¯# h

τ¯♭ ≡ g♭ (¯ τ ) ∈ H∗

τ U

¯ U⋆ h ∈ U⋆ ∨ # $ # $ zA ≡ z1 , z2

h = ¯z1 ⊗ z1 + ¯z2 ⊗ z2 ¯#

h

ϵ# (z1 ) = z2 ϵ# (z2 ) = −z1 = 2 τ0 √

¯ # = √2 τ0 h

⇒

h# = ¯z1 ⊗ z1 + ¯z2 ⊗ z2 # $ τλ

h W # $ ¯ ′ ) ≡ h(¯ ¯ ′ , λ) . h (¯ u, λ), (u′ , λ u, u′ ) + h# (λ

¯ ψ ′ ) = k(γ0 ψ, ψ ′ ) = k(ψ, ¯ γ0 ψ ′ ) , h(ψ,

γ0 ≡ γ[τ0 ] ≡

√1 2

ψ∈W ¯ = k♭ (γ0 ψ) = k♭ (ψ) ¯ ◦ γ0 ∈ W ⋆ , ψ † ≡ h♭ (ψ)

γ0 ◦ γ0 = g00 11 = 11 ¯ = h♭ (ψ) ¯ ◦ γ0 ≡ ψ † ◦ γ0 . ψ¯♭ ≡ k♭ (ψ) ψ¯ = ψ † γ0 ψ¯ ψ† ψ ¯ ψ (→ h♭ (ψ)

h ψ † γ0

W

h

γ0

¯#] . γ[h

h=

√

2 τ¯0

τ0 h

W

h

¯ (→ pψ ≡ W ≡ U ⊕ U ⋆ → H ∗ : ψ ≡ (u, λ)

1 √ 2

¯ , (u♭ ⊗ u ¯♭ + λ ⊗ λ) ψ

pψ ∈ H ∗

# p# ψ ≡ g (pψ ) =

√1 2

¯#) ∈ H , (u ⊗ u ¯ + λ# ⊗ λ

γ[p] ≡ γ[p# ] * + ∗ ¯ µ ¯ . ¯ (λ, γ[pψ ](v, µ ¯) = ⟨¯ µ, u ¯⟩ u + ϵ ¯) λ# , ϵ(u, v) u ¯♭ + ⟨λ, v⟩ λ ¯ ≡ (v, µ ψ ≡ (u, λ) ¯)

# $ ¯ = ⟨λ, ¯ u ¯ . γ[pψ ](u, λ) ¯⟩ u , ⟨λ, u⟩ λ

# $ ¯ u ¯ , γ[p]ψ = ⟨λ, ¯⟩ u , ⟨λ, u⟩ λ

¯ , ψ ≡ (u, λ)

p ∈ H∗ ¯ ∈W ψ ≡ (u, λ)

p = pψ

γ[p]ψ = ±m ψ , 5

p ∈ H∗ ⟨λ, u⟩ = ±m , p = pψ ,

m ≡ (g# (p, p))1/2 = |⟨λ, u⟩| ∈ {0} ∪ R+ m ∈ L−1

p ∈ H∗

γ[p] ◦ γ[p] = m2 11W W = Wp+ ⊕ Wp− ,

Wp±

±m

γ[p] ψ = 12 (ψ +

1 m

γ[p]ψ) + 12 (ψ −

1 m

γ[p]ψ) .

% & ¯ ∈ W : ⟨λ, u⟩ = ±m ⊂ W , W ± ≡ (u, λ) γ[p]

W

ψ¯♭ (→ ψ¯♭ ◦ γ[p]

W

W±

C:W →W γ[p]

C

ψ ∈ Wp±

γ[p]Cψ = −C γ[p]ψ = ∓Cψ . #1

u, m

p/m

1 # λ m

$

ψ U

H

τ0 γ 0 ≡ γ [ τ0 ]

γη

P ≡ γ0

C T ≡ γη ◦ γ0 ◦ C ≡ γη ◦ P ◦ C .

P 2 = 11 ,

T 2 = −11 ,

PC = CP = γ0 C ,

PT = T P = −γη C ,

CT = −T C = γ0 γη ,

(CT )2 = −(PC)2 = −(PT )2 = 11 , T

P

PCT = γη ,

X

γ′ [w] ≡ X γ[w]X −1 ,

′

′

γ [w]ψ = ±m ψ

w∈H, ψ′ ≡ X ψ

γ[w]ψ = ±m ψ

′

C γ[w]C −1 = PCT γ[w](PCT )−1 = −γ[w] = −PT γ[w](PT )−1 , P γ[w]P −1 = T γ[w]T −1 = 2g(w, τ0 )γ0 − γ[w]

= −PC γ[w](PC)−1 = −CT γ[w](CT )−1 .

τ0 j : U ⊗ U → U ⊗ U ⋆ : u ⊗ v¯ (→ u ⊗ (τ♭0 ⌋v¯) , U

τ♭0 ≡ g ♭ (τ0 ) ∈ U ⋆ ⊗ U ⋆

j

ji ≡ j[τi ] = ˆ iAB ≡ σiAA˙ σ0BA˙ σ

1 2

ˆ iAB zA ⊗ zB , σ

# $ σi

# $ ˆi σ

j = ji τi : H⊥ → EndU , % & H⊥ ≡ w ∈ H : g(τ0 , w) = 0 ⊂ H τ0

# $ J ≡ j , ¯j∗ : H⊥ → EndW ,

J[w] =

i 2

γ0 γ[w] γη . Ji ≡ J[τi ]

# $ Ji =

6

( σi ) (0)

(0) ( σi )

7

.

H⊥

−g

J (H⊥ , ×)

−iJ

(EndW , [ , ]) ∗

−i J ≡ iJ∗ : H⊥ → EndW ⋆ ,

n ∈ H⊥

# $ ∗ J ≡ −J∗ = −j∗ , −¯j . g(n, n) = −1

n

J[n] = ni Ji

J[n] ◦ J[n] = 14 11W . ± 12

J[n] W 1 11 2

n

± J[n] −gij Ji ◦ Jj =

¯ ∈W ψ ≡ (u, λ) p

3 4

11W ,

p ≡ pψ ≡

p⊥ ≡ p − ⟨p, τ0 ⟩ τ0 = p − nψ ≡

1 |p|

1 2

√1 2

¯ (u♭ ⊗ u ¯♭ + λ ⊗ λ)

# $ ⊥∗ ¯ λ) τ0 ∈ H ∗⊥ ∼ h(¯ u, u) + h# (λ, =H ,

(p⊥ )# ∈ H⊥ ,

J[nψ ] ∈ EndW

√

0 2τ

¯. =h

|p| ≡ |g# (p⊥ , p⊥ )|1/2 . ψ

¯ ∈W ψ ≡ (u, λ)

λ# ̸∝ u λ# ≡ ϵ# (λ) c ∈ C\{0} |c| ̸= 1 ¯ = cu . h# (λ) 1 2

pψ ψ

u

J[n(u,0) ](u, 0) = (u, 0)

1 2

(u, 0) , ¯ (0, λ)

sgn(|c|2 − 1) λ# ∝ u λ

¯ = − 1 (0, λ) ¯ , J[n(0,λ) ¯ ](0, λ) 2

ψ

U ≡ EndU ≡ U ⊗ U ⋆ ,

H ≡ EndH ≡ H ⊗ H ∗ ,

U = S ⊕ C11U = S ⊕ R11U ⊕ iR11U , H = G ⊕ R11H ⊕ G′ , • S⊂U

U

• G⊂H

• G′ ⊂ H • R11

H

g

H

g C11 Ξ∈H Ξ=

1 2

#

$ Ξ − Ξ† +

1 4

TrΞ 11H +

p : H → G ⊕ R11H : Ξ (→

#1

1 2

H

2

#

(Ξ + Ξ† ) −

$ Ξ − Ξ† +

1 4

1 4

$ TrΞ 11H .

TrΞ 11H ,

G′

ξ∈U ξ = (ξ − 12 Trξ 11U ) + 12 Trξ 11U ,

R

H ⊂ U ⊗U

π:H→U,

ι:U→H.

g Ξ∈H g Ξ† ≡ g♭ ◦ Ξ∗ ◦ g# ∈ H ι

Ξ† = Ξ Ξ† = −Ξ (Ξ† )λµ = gλν Ξρν gρµ

g ! " ιµ

(πΞ)AB =

1 2

˙ ΞAABA ˙−

1 8

˙ A ΞCC CC ˙δ B .

ιξ ≡ ξ ⊗ 1¯1 + 11 ⊗ ξ¯ ˙ A A˙ A ¯A˙ (ιξ)AABB ˙ = ξ B δ B˙ + δ B ξ B˙ .

• π11H =

1 2

11U ι11U = 2 11H

• ker π = ker p = G′ • ker ι = iR11U •

p

G ⊕ R11H

ι

•

G ⊕ R11H

π

ι

S ⊕ R11U

R α : U ≡ EndU → EndW : ξ (→ α(ξ) ≡ (ξ, −ξ¯∗ ) ¯ = (ξ(u), −λ ¯ ◦ ξ) ¯ , α(ξ)(u, λ)

⇒

ξ¯∗ : U ⋆ → U ⋆ U, U⋆ ⊂ W γ(∧H)

ξ

α

α(ξ) : U −→ iR 11W ⊕ γ[∧2 H] ⊕ iRγη ⊂ EndW . α ◦ π : H → EndW α(π Ξ) = 18 Ξ[λµ] γλ γµ − 8i Ξνν γη ,

H

π

p

S ⊕ R11U

α

γ[∧2 H] ⊕ iRγη

ι

G ⊕ R11H

ι

ιξ ιξ(u ⊗ u ¯) = ξu ⊗ u ¯ + u ⊗ ξ¯u ¯

u⊗u ¯∈H

.

.

. . .

.

.

S!M M S!M

S

C

M

L!M U !M Q!M H!M W !M # a$ x # $ # $ # $ (ll) zA (ϵ) τλ ζα − Γ S→M − (xa , bA ) ΓaAB : M → C s:M →S

# $ bA

∇s = (∂a sA − −ΓaAB sB ) dxa ⊗ bA . ¯ Γ

−

∗ −

Γ

¯ Γ

− A˙ a B˙

= −ΓaAB ,

S!M S⋆ ! M ∗ −

ΓaAB = −−ΓaBA ,

∇¯ s = ∇s ¯ S⋆ ! M Γ

∗ −

∗ −

˙ ¯aAB ¯ B˙ − Γ ˙ = −Γa A˙ .

S Γ

−

Λ!M

cr ≡ ⊗rc (cr )a = rca

c Λr ≡ Λ ⊗ ··· ⊗ Λ r ∈ N

r∈Q

Gauge Field Theory in Natural Geometric Language: A revisitation of mathematical notions of quantum physics. Daniel Canarutto, Oxford University Press (2020). © Daniel Canarutto. DOI: 10.1093/oso/9780198861492.001.0001

S 2

∧S

−

Γ

EndS

− A a B

ˆ Γ

C

−

−

L

G

U

−

Q ≡ ∧2 U

Y

2

∗ ˜≡G Γ ⊗ −Γ

2

H

− D

∗

R

2 Ga

U ≡ EndU

− A a B

End(S ⊗ S)

˙ ¯ aAABB (−Γ ⊗ −Γ) ˙

2 Ga G

⊗Y

ˆa 2Ya = ℑ−Γ ˙ ˜aAABB Γ ˙

iR 11W ⊕ γ[∧2 H]

L!M 2

˜ Γ

G ⊕ R11H ⊂ H

∗ − −

¯ Γ ≡ G ⊗ (Γ, Γ)

W

ˆa Γ

iR

¯ Γ′ ≡ −Γ ⊗ −Γ ¯ ˜ ≡ −Γ ˜ ⊗ −Γ ˜ Γ

S ⊗S

Γ

2iYa

G

(−ΓD )aαβ

∧2 U ! M

G

2

Y

2

L1/2 ! M

∧S!M

2

2

∇ll = −2 Ga dxa ⊗ ll ,

∇ϵ = 2 iYa dxa ⊗ ϵ ,

∇(ll−1 ⊗ ϵ) = 2(Ga + iYa ) dxa ⊗ ll−1 ⊗ ϵ .

Ga

Ya

Ga

˙ ¯aAA = ℜ( 12 −ΓaAA ) = 14 (−ΓaAA + −Γ ˙) ,

Ya

= ℑ( 12 −ΓaAA ) =

1 − A (Γa A 4i

˙ ¯aAA − −Γ ˙) .

2G ∧2 U ! M

U ≡ L−1/2 ⊗ S ! M # $ # −1/2 $ zA ≡ ll ⊗ bA ˜ Γ

− A a B

= −ΓaAB − Ga δAB .

¯ Γ′ ≡ −Γ ⊗ −Γ

S ⊗S ! M ,

¯ ˜ ≡ −Γ ˜ ⊗ −Γ ˜ Γ

U ⊗U ! M ,

L!M

∗ ˜≡G Γ ⊗ −Γ

−

2Y

˙ A˙ A − ¯ A˙ − A Γa′AABB ˙ = Γa B δ B˙ + δ B Γa B˙ ,

¯ A˙ ˙ A − ′ AA˙ A A˙ ˜aAABB ˜ A A˙ ˜ − Γ ˙ = Γa B δ B˙ + δ B Γa B˙ = Γa BB˙ − 2 Ga δ B δ B˙ ˙ A − A A˙ ¯ A˙ = −ΓaAB δAB ˙ + δ B Γa B˙ − 2 Ga δ B δ B˙ .

˜ Γ′ Γ ¯ H ≡ U ∨U ! M

¯S !M S∨

g

H

¯⊗ϵ ϵ

∇(¯ ϵ ⊗ ϵ) = 0 ˜ g=0, ∇[Γ]

H!M ˜ Γ ˙ λ BB˙ ˜aλµ = Γ ˜aAABB Γaλµ ≡ Γ ≡ ˙ τ AA˙ τµ

G ⊂ End H Γaλµ

≡

Γaλν

g

νµ

Γaµµ

g

˙ ¯ aAABA (−Γ ⊗ −Γ) ˙

Γ

−

− A a B

Γ

˜ R

=0

˙ ΓaAABA ˙

= (−Ga + iYa ) δAB + = (Ga + iYa ) δAB +

Ω

˙ λ BB˙ ˜aAABB Γ ˙ σ AA˙ σµ

H!M

M → T∗M ⊗ G g Γaλµ

1 2

Γ

−

1 2

Γaλµ + Γaµλ = 0 ∇η = 0

˙ 1 − ¯ aAABA (Γ ⊗ −Γ) ˙ 2 ˙ ˜aAABA Γ ˙.

˜ Γ

ΩabAB = −2 (dG + i dY )ab δAB +

1 2

˙ ˜ abAABA R ˙,

dG G

Y

dY M

L!M

dG = 0 Ga = 0

G

L!M

L

∗

∗

¯ Γ ≡ G ⊗ (−Γ , −Γ)

− D

W ≡ U ⊕ U⋆ ! M M

Γaαβ ≡ (−ΓD )aαβ = iYa δαβ +

−

1 4

Γaλµ (γλ γµ )αβ ,

Ωabαβ ≡ (ΩD )abαβ = −2i dYab δαβ +

1 4

˜ λµ (γλ γµ )αβ , R ab

ΩD ≡ −[[−ΓD , −ΓD ]] Γaλµ =

1 4

Tr(γ[λ (−ΓD )a γµ] ) ,

C

ˆ ≡ Tr −Γ , Γ

Γ

−

1 4

Tr(γ[λ Ωab γµ] ) ,

S!M

L!M

M → T∗M ⊗ L −

˜ λµ = R ab

Γ′ ≡ ι−Γ ,

˜ Γ

− A a B

˜ ≡ ι−Γ ˜, Γ

= π(ι−Γ)aAB = =

1 2

1 2

˜ , Γ = Y ⊗ α(−Γ)

− D

˙ (ι−Γ)aAABA ˙−

1 8

˜ = π(ι−Γ) = π(ι−Γ) ˜ . Γ

−

˙ A (ι−Γ)aCC CC ˙δ B

˙ ˜aAABA ˜ A − Γ ˙ = π(ιΓ)a B .

M M

θ : TM → L ⊗ H . θ H!M

M

L

H!M

S!M

M

θ : M → L ⊗ H ⊗ T∗M . θ θ ←

θ : L ⊗ H → TM ,

←

θ : M → L−1 ⊗ TM ⊗ H ∗ . H

g ◦ (θ, θ) : M → L2 ⊗ T∗M ⊗ T∗M , M

#

τλ

$

L−1 ⊗ TM

g # $ g(v, v ′ ) ≡ g θ(v), θ(v ′ )

#

v, v ′ ∈ TM

$ #← $ θ λ ≡ θ ( τλ )

˙ θ = θλa τλ ⊗ dxa = θAA zA˙ ⊗ dxa , a zA ⊗ ¯

θλa : M → R ⊗ L , ˙ θAA = τλAA˙ θλa a M

˙ θAA : M → C⊗L , a ˙A ˙ ¯A θ = θAA a a

˙ BB˙ ¯A˙B˙ θAA g = gab dxa ⊗ dxb ≡ gλµ θλa θµb dxa ⊗ dxb = ϵAB ϵ dxa ⊗ dxb , a θb

gλµ = 2δ0λ δ0µ − δλµ

θ gab ≡ gλµ θλa θµb : M → R ⊗ L2

θ 4

η:M →∧H

∗

g M

θ∗ η

θ∗ η : M → L4 ⊗ ∧4 T∗M H!M θ η γ : H → EndW

γ ◦ θ : TM → L ⊗ EndW , γ η = det θ d4 x = | det g|1/2 d4 x , γ = θλa dxa ⊗ γλ =

√ AA˙ a 2 θa dx

¯A˙B˙ ¯zB˙ ⊗ zB ) , ⊗ (zA ⊗ ¯zA˙ + ϵAB ϵ

d4 x ≡ dx1 ∧ dx2 ∧ dx3 ∧ dx4 ←

θ = θaλ ∂xa ⊗ τλ ,

θaλ : M → R ⊗ L−1 ,

#← $ θaλ ≡ θ aλ

θλa

θaλ = gab θµb gµλ . #

θ

# λ$ τ

$ λ

#

≡ θ ∗ ( τλ ) ←

$

θλ ≡ θ (τλ ) = θaλ ∂xa ,

#

$ #← $ θλ ≡ θ (τλ )

θλ ≡ θ∗ τλ = θλa dxa ,

U !M

θ

←

θ

λ = 0, 1, 2, 3

!

xa

"

M

ξ : M → ∧p T∗M ⊗ ∧p H ,

ξ ′ : M → ∧q T∗M ⊗ ∧q H ,

ξ ∧ ξ ′ : M → ∧p+q T∗M ⊗ ∧p+q H ,

(α ⊗ w) ∧ (α′ ⊗ w′ ) ≡ (α ∧ α′ ) ⊗ (w ∧ w′ ) .

∧q θ : M → Lq ⊗ ∧q T∗M ⊗ ∧q H , ∧q θ : ∧q TM → Lq ⊗ ∧q H ∧q θ(u1 ∧ · · · ∧ uq ) ≡ θ(u1 ) ∧ · · · ∧ θ(uq ) ,

η

H

∧q θ

q = 1, 2, 3, 4 .

∧q H → ∧4−q H ∗ ˘[q] : M → Lq ⊗ ∧q T∗M ⊗ ∧4−q H ∗ . θ ˘[q] θ

dxa1 ...aq ≡ (∂xa1 ∧ · · · ∧ ∂xaq )|d4 x

(

).

˘[4] = θ∗ η = |θ| d4 x : M → L4 ⊗ ∧4 T∗M , θ ˘[3] ≡ θ ˘=θ ˘aλ dxa ⊗ τλ : M → L3 ⊗ ∧3 T∗M ⊗ H ∗ , θ λ µ 2 2 ∗ 2 ∗ ˘[2] = θ ˘ab θ λµ dxab ⊗ τ ∧ τ : M → L ⊗ ∧ T M ⊗ ∧ H , λ µ ν ∗ 3 ∗ ˘[1] = θ ˘abc θ λµν dxabc ⊗ τ ∧ τ ∧ τ : M → L ⊗ T M ⊗ ∧ H ,

| θ| ≡

1 4!

ϵabcd ϵλµνρ θλa θµb θνc θρd = det θ ,

˘aλ ≡ θ

1 3!

ϵabcd ϵλµνρ θµb θνc θρd =

∂|θ| , ∂ θλa

˘ab θ λµ =

1 2

ϵabcd ϵλµνρ θνc θρd =

˘aλ θλa = 4 |θ| , θ

˘ab ∂θ λµ abcd ˘abc θ ϵλµνρ θρd = . λµν = ϵ ∂ θνc

˘aλ ∂θ , ∂ θµ b

λ ˘ab ˘b θ λµ θa = 3 θµ ,

λ ˘abc ˘b ˘c θ λµν θa = 2 θµ θν .

θ ˘ab θ λµ =

1 | θ|

←

←

←

←

˘aλ θ ˘bµ − θ ˘aµ θ ˘bλ ) = |θ| ( θ aλ θ bµ − θ aµ θ bλ ) , (θ ←

←

←

←

a b c ˘abc θ λµν = |θ| θ [λ θ µ θ ν] ,

θ aλ =

1 | θ|

←

← ∂| θ | ˘cν . = −| θ |2 θ ν ∂ θc

˘aλ , θ

˘≡θ ˘[3] : M → L3 ⊗ ∧3 T∗M ⊗ H ∗ , θ

3 3 ∗ ˘ : H → L3 ⊗ TM ⊗ ∧4 T∗M ∼ θ = L ⊗∧ T M ,

˘ θ

θ

θ

η ≡ θ∗ η

θ

M ←

˘ = θ ⊗η . θ

M TM ! M

(θ, −Γ) S!M ˜ Γ

− H!M Γ TM ! M θ : M → L ⊗ T∗M ⊗ H

c λ ∇a θλb = ∂a θλb − Γaλµ θµ b + Γa b θc ,

θ

Γ

˜aλµ . Γaλµ ≡ Γ ∂a θλb

L 2G

G

˜ Γ

∇θ = 0 ˜aλµ Γaλµ ≡ Γ

Γ

θ Γ

Γ

# $ Γacb = θcλ −∂a θλb + Γaλµ θµ b ,

←

# $ #← $ θ ≡ θ (τλ ) # aλ$ x

θcλ ≡ θ cλ .

˜ Γ

Γ

˜ abλµ θcλ θµ = (Γ λ + Γ λ Γ ν ) θcλ θµ . Rabcd = R d d [a,b] µ [a ν b] µ

T : M → TM ⊗ ∧2 T∗M

Γ

θ⌋T = [[Γ, θ]] .

# λ $ T cab = −Γacb + Γbca = θcλ ∂[a θλb] + θµ[a Γb] µ . T˘ : M → T∗M

˘aAA˙ T bab = [[Γ, ˘]]AA˙ = ∂a θ ˘aAA˙ + ˜ θ TAA˙ ≡ θ

Γ

1 2

T˘a = T bab

˙ CB˙ ˘aBA˙ ΓaBC AC ˘a (θ ˙ + θAB˙ Γa CA˙ ) .

(θ, −Γ)

Γ

R

[[θ, R]] = [[Γ, θ⌋T ]] ,

0 = [[θ, [[Γ, Γ]]]] + [[Γ, [[Γ, θ]]]] + [[Γ, [[θ, Γ]]]] ,

(θ, Γ) −

∇

[[Γ, R]] = 0

0 = [[Γ, [[Γ, Γ]]]] ≡ −[[Γ, R]] .

¯ :M →W ψ ≡ (u, λ)

←

θ

˘ θ

θ ∇ψ ≡ ∇[−Γ]ψ : M → T∗M ⊗ W ,

˘ : M → L3 ⊗ H ∗ ⊗ TM ⊗ ∧4 T∗M ; θ

˘ ⊗ γ ⊗ ∇ψ θ

γ : H → EndW

˘ : M → L3 ⊗ W ⊗ ∧4 T∗M , ∇ψ

˘ =θ ˘aλ γλαβ ∇a ψ β d4 x ⊗ ζα ∇ψ

# $ ¯ A˙ zA , σλAA˙∇a uA ¯zA˙ . = d4 x ⊗ σλAA˙ ∇a λ ←

θ

←

θ ⊗ γ ⊗ ∇ψ

˘ θ ∇ψ : M → L−1 ⊗ W

˘ = ∇ψ ⊗ η , ∇ψ

∇ψ = gλµ θaλ γµαβ ∇a ψ β ζα . 2

∇ ≡∇◦∇ 2

←

θ

γ

←

∇ ψ = θ aλ θ bµ γλ γµ ∇a ∇b ψ ≡ γa γb ∇a ∇b ψ = (γa ∧ γb + gab 11) ∇a ∇b ψ = "ψ + γa ∧ γb ∇a ∇b ψ = "ψ +

1 2

γa γb (ΩD )ab ψ ,

"ψ

ΩD

R

Y

2

∇ ψ − "ψ =

1 4

⟨ R⟩ ψ # $ + 12 γλ ∧ γµ Ricλµ − ϵabcd θdλ [[Γ, θ⌋T ]]abcλ γη − i γa γb dYab ψ . Y

=0

Γ

Γ

θ

−

T

Ga

=0 T ⊂M

TT ⊂ TM τ : T → L−1 ⊗ TT ⊂ L−1 ⊗ TM

UT ! T TT M ! T HT ! T

θ ◦ τ : T → HT , τ ˜ Γ

Γ

−

UT ! T

∼ L⊗H TM =

Γ

− T

HT ! T

ΓT

∗ ∗ Φ : M → T∗ T ⊗ EndHT ∼ = T T ⊗ HT ⊗ HT , T

T

v ⌋Φ ≡ (∇v τ ) ⊗ τ ♭ − τ ⊗ (∇v τ )♭ : w (→ g(τ, w) ∇v τ − g(∇v τ, w) τ , v ∈ TT # $ τλ

w ∈ HT τ0 ≡ τ

T

τ0

T T

# λ$ τ

˙ 0 Φ = Φ0λµ τ0 ⊗ τλ ⊗ τµ ≡ Φ0AABB zA˙ ⊗ zB ⊗ ¯zB˙ , ˙ τ ⊗ zA ⊗ ¯ ˙ λ AA˙ µ Φ0AABB τ BB˙ ≡ ˙ = Φ0 µ τλ

1 2

Φ0λµ σλAA˙ σµBB˙ ,

(v ⌋Φ)λµ = v 0 Φ0λµ v ⌋Φ

˙ Φ0λλ = Φ0AAAA ˙= 0

g Φ

φ ≡ πΦ : T → T∗ T ⊗ EndUT , T

φ = φ0AB τ0 ⊗ zA ⊗ zB ≡

1 2

˙ 0 B Φ0AABA ˙ τ ⊗ zA ⊗ z .

Φ = ιφ ˙ A A˙ A ¯ A˙ Φ0AABB ˙ = φ0 B δ B˙ + δ B φ0 B˙ .

w

w : T → HT

∇v w ≡ ∇v w ˜

w ˜

φ0AB =

1 2

˜ 0j σ A . Γ 0 j B

˙ φ0AA = Φ0AAAA ˙= 0 φ UT ! T Φ HT ! T

v ⌋φ

Γ

−

˜ Γ

˜+Φ, ΓF ≡ Γ

Γ ≡ −Γ + φ ,

− F

Γ

− F

ΓF

T ¯F Γ ⊗ −Γ

− F

i α ⊗ 11

Γ

− F

HT ! T − ΓF′ UT ! T ∗ α:T →T T Γ

− F

ΓF

ΓF

(−ΓF′ )0AB = (−ΓF )0AB + i α0 δAB .

M ΓF α=0 u : T → UT D u ≡ ∇0 [−ΓF ]u ,

w : T → HT

D w ≡ ∇0 [ΓF ]w .

D uA = ∇0 uA − φ0AB uB = ∂0 uA − −Γ0AB uB −

$ ← ←# D θ (w) ≡ θ D θ(w) Dg=0

1 2

˜ 0j σ A uB . Γ 0 j B

TT M ! T

HT = HT∥ ⊕ H⊥ T T

τ Dτ =0 − ΓF ∗ ¯ − Γ ≡ (−Γ, −Γ)

τ Γ

− F

WT ! T ∗

∗

¯ ∗) . ¯F ) = (−Γ+φ , −Γ− ¯ φ Γ ≡ (−ΓF , −Γ

− F

¯ ∗) = (φ, −φ

1 4

ˆ (Φ) = γ

1 4

˜ 0j dx4 ⊗ (γ0 γj − γj γ0 ) , Γ 4

ˆ : ∧H → EndW γ (−ΓF )0αβ = −Γ0αβ +

1 4

˜ 0j (γ0 γj − γj γ0 )αβ Γ 0

= iYa δαβ +

1 4

Γaλµ (γλ γµ )αβ +

1 4

˜ 0j (γ0 γj − γj γ0 )αβ . Γ 0

p : T → H∗ WT = Wp+ ⊕ Wp− ! T . ¯ : T → WT ψ ≡ (u, λ)

ψ ± : T → Wp± ¯ : T → WT ψ ≡ (u, λ) p≡

√1 2

¯ (u♭ ⊗ u ¯♭ + λ ⊗ λ)

M

d(x, y) = sup ∥∇f ∥≤1

%

& f (y) − f (x) ,

x, y ∈ M .

% & d(x, y) = sup |f (y) − f (x)| : ∥[D, f ]∥ ≤ 1 , f ∈A

1

A ≡ C (M , R) D

M M

# $ A, H, D ,

H

A

D

d (x, y) C1

x

y

8 9 d (x, y) = inf [f (x) − f (y)]+ , f ∈S

df

S

W !M

x

y

[r]+ ≡ max{0, r} , r ∈ R , M →R

⟨df, τ ⟩ ≥ ∥τ ∥

• •

d (x, y) = 0

A ≡ C1 (M , R) H∼ = L2 (M , W )

h

W

∇ ≡ −i θaλ γλ ∇a ≡ −i γa ∇a

•

iγ0

• •

χ ≡ ±iγη = ±iγ0 γ1 γ2 γ3

d (x, y) = inf

f ∈A

8

9 # $ [f (x) − f (y)]+ : h ψ¯ , γ0 ([∇, f ] ± γη )ψ ≥ 0 ∀ψ ∈ H . γ0

τ0

h ¯ γ0 φ) = k(ψ, ¯ φ) h(ψ,

d (x, y) = inf

f ∈A

8

k

9 # $ [f (x) − f (y)]+ : k ψ¯ , ([∇, f ] ± γη )ψ ≥ 0 ∀ψ ∈ H .

τ

H

h

[∇, f ]ψ = ∇(f ψ) − f ∇ψ = γa ∇a (f ψ) − f ∇ψ = (γa ∂a f ) ψ = γ# [df ]ψ ≡ /df ψ , ¯ : M → U ⊕ U⋆ ≡ W ψ ≡ (u, λ)

# $ # $ k ψ¯ , ([∇, f ] ± γη )ψ = k ψ¯ , (/df ± γη )ψ = ∓2 |⟨λ, u⟩| sin α +

α ≡ arg ⟨λ, u⟩ sin α =

τ ≡

√ :← ∗ ;$ ¯ # , 2 θ (df ) , u ⊗ u ¯ + (λ ⊗ λ) ¯ u ⟨λ, u⟩ − ⟨λ, ¯⟩ . 2 i |⟨λ, u⟩|

¯♭) , (u ⊗ u ¯ + λ♭ ⊗ λ

√1 2

|τ |2 ≡ g(τ, τ ) = |⟨λ, u⟩|2 ,

# $ k ψ¯ , (/df ± γη )ψ = 2 ⟨df, τ ⟩ ∓ 2 |τ | sin α .

d (x, y) = inf

f ∈A

I α

+

⊂H

8

τ ∈H

¯ ψ ≡ (u, λ)

9 [f (x) − f (y)]+ : ⟨df, τ ⟩ ∓ 2 |τ | sin α ≥ 0 ∀τ ∈ I + , α ∈ R ,

τ ∈ I+ ⟨df, τ ⟩ ≥ |τ | sin α ⟨df, τ ⟩ ≥ −|τ | sin α

∀α ∈ R , ∀α ∈ R ,

⟨df, τ ⟩ ≥ |τ | .

k

γη

d (x, y) = inf

f ∈A

8

9 [f (x) − f (y)]+ : ⟨df, τ ⟩ − |τ | ≥ 0 ∀p ∈ I + , ψ ψ

.

.

. . .

.

.

p:F !B φ:B→F m L : JF → ∧m T∗B ⊂ ∧m T∗ JF , L = ℓ dm x

m ≡ dim B .

ℓ : JF → R r1 : JTF → TJF

v = v a ∂xa + v i ∂yi : JF → TF F J2 F ⊂ JJF

Jv : JJF → JTF

v(1) ≡ r1 ◦ Jv = v a ∂xa + v i ∂yi + vai ∂yia : J2 F → TJF , j j vai = da v i − da v b ybi = (∂a v i + ∂j v i yaj + ∂jb v i yab ) − (∂a v b + ∂j v b yaj + ∂jc v b yac ) ybi .

TF ̸⊂ TJF

v L[v]L ≡ d(v|L) + v(1) |dL : J2 F → ∧m T∗JF

⇒

# $ L[v]L = ∂a (ℓ v a ) + ∂i ℓ v i + ∂ia ℓ (da v i − da v b ybi ) dm x + ℓ (∂i v a dyi + ∂ib v a dybi ) ∧ dxa .

Gauge Field Theory in Natural Geometric Language: A revisitation of mathematical notions of quantum physics. Daniel Canarutto, Oxford University Press (2020). © Daniel Canarutto. DOI: 10.1093/oso/9780198861492.001.0001

L

B

v

δ[v]L ◦ j2 φ = jφ∗ L[v]L

δ[v]L : J2 F → ∧m T∗B

∀φ : B → F .

va = 0

v

L[v]L = δ[v]L = (∂i ℓ v i + ∂ia ℓ da v i ) dm x # $ = (∂i ℓ − da ∂ia ℓ) v i + da (v i ∂ia ℓ) dm x ˇ ⌋v) : J2 F → ∧m T∗B , ≡ E ⌋v + dH (dL E : J2 F → ∧m T∗B ⊗F V∗F ˇ = ∂ia ℓ dxa ⊗ dyi : JF → ∧m−1 T∗B ⊗ V∗F , dL L φ:B→F

C⊂B

←

v : F → VF