Algebraic QFT & Local Gauge Invariance (Mathematical Aspects of Quantum Fields and Related Topics)

Algebraic QFT

& Local

_Gauge

Invariance

Iztuni

OJIMA

1

What is aimed

at

here

It has been common to _{discuss local gauge invariance in close relation with}

indefinite inner

_product

of a state vector space which violates the basic

hypotheses

for

_{probabilistic}

_{interpretation}

in

_QFT.

In

_sharp

_contrast,

such

basic structures as the

validity

of Malwell

equation

can be determined

algebraically

and/or

categorically

in

_{algebraic QFT}

withoutreferencetothe

concept

of_{state vector spaces}

_(with

or withoutindefiniteinner

products)

as seen below.

1.1

_Quadrality

scheme related with

5\mathrm{W}\mathrm{I}\mathrm{H}

Tothis

_end,

we start our discussion here with ourtheoretical framework in

thecontextof Micro‐Macro

_duality

in

_quadrality

schemeas follows. For the

purpose of

ensuring

1)

bi‐directionality

in inductive 8 deductive

_arguments,

we need aframeworkto accommodate induction processes

theoretically

\Rightarrow \mathrm{a}

_possible

candidate for such a universal theoretical frameworkcan Ue

found in

_quadrality

scheme: ,

consisting

of the

_following

₄₍₊₁₎

basic

_{ingredients adapted}

to “5WIH”:

Spec

[When

& Where=

classifying

_{space to}

specify

events],

State\grave{s}

_[Who

=

“subject”

to

specify

context],

Alg(ebra

of

_{variables) [What}

=

objects

to be

described],

(Rep

(of Alg)) [How

= modus of

phenomena

=

“modules”],

1.2

_Emergence

&

_quantum

fields in

_quadrality

scheme

2)

Bi‐directional relations becomeeffectivebetween

_{phenomenogical}

visible

1.3

_Quadrality

scheme

combined with

various

dualities

₍₁₎

3)

Combined with

horizontal

_{Fourier‐type dualities,}

.

States^{\leftarrow}\rightarrow Rep\rightarrow\leftarrow Alg,

due to

_operator

_algebra

_theory,

two non‐trivial

_ingredients

in the

_scheme,

physical

emergence

of

Spec from

States

&

quantum

fields

on

emergent

Spec,

entail the

_following

network

_of

connections over this

quadrality

scheme:

1.4

_Quadrality

scheme combined with various

duaIities

₍₂₎

The actual

_meaning

of

_Spec=

_[When

&

_{\mathrm{W}\mathrm{h}\mathrm{e}\mathrm{r}\mathrm{e}|}

to

_specify

event local‐ izations is ensured

_by

its

_origin

_{of the emergence processes,}

_physically

as

phase separations,

and

_{mathematically}

_{by forcing}

method of

_identify‐

ing

the extended semantic space

(of

multi‐valued

_logic).

_Owing

to

_duality

(Alg)^{*}=

States,

(States)

=Alg

, the emergence arrow, States

\rightarrow Spec,

implies

the dual arrow of co‐emergence to create

_objects

in

_Alg

from the

1.5

_Quadrality

scheme

combined with various dualities

₍₃₎

In the

_opposite

_direction,

local

_net,

_{Spe\mathrm{c}\rightarrow Alg}

of

_quantum

_fields

_triggers

_{induction processes, in‐}

ducing

five downward arrows, among which Galois _{functor identifies group}

in

_Dyn

from the

_{representation}

contents

_Rep.

2

_Emergence

of

_Spec

as

sector‐classifying

_space

Inthisway,wehave learned thecrucialrolesof_{emergence process in}

_creating

Spec

via

_Bottom‐Up.

_{This process} can be formulated as follows in terms

of the

_concepts

of\cdot

sectors in the case of

QFT:

1)

Sectors=pure

phases

parametrized

by

order

parameter[

=_macro‐

scopic

centralobservables

_{3_{ $\pi$}(\mathcal{X})= $\pi$(X)\cap $\pi$(\mathcal{X})}

_commutilB

with all

_phys‐

ical variables

_{$\pi$(\mathcal{X})}

in a

generic

_{representation}

$\pi$ of

algebra

\mathcal{X} of

phys‐

ical

_variables]:

_{mathematically,}

a sector

(=pure

phase

)

\mathrm{d}\mathrm{e}\mathrm{f}=

a

quasi‐

equivalence

class

_{of factor}

states

_(&

_{representations}

_{$\pi$_{ $\gamma$}}

₎

of

_(C

*

‐)algebra

\mathcal{X} of

_{physical variables,}

as a minimal unit of

representations

characterized

by

trivial centre

_{$\pi$_{ $\gamma$}(\mathcal{X})^{n}\cap$\pi$_{ $\gamma$}(\mathcal{X})=:3_{$\pi$_{ $\gamma$}}(\mathcal{X})=\mathbb{C}1.}

*)

Important

remark: in the usual

_quantum

mechanics with

_{finite degrees}

offreedom,

sectors are

replaced by

irreducible

_{representations}

& _pure states

with

_Spec=

_{

one

point}!

_They

become

_meaningless,

_however,

in the

_general

contexts

_involving

_quantum

fieldswith

_{infinite degrees offreedom}

which

_play

crucial roles in

_connecting

invisible Micro and visible Macro.

2.1

Micro‐Macro

_Duality

of

Intra‐

vs. Inter‐sectorial levels

2)

The roles of sectors as Micro‐Macro

boundary:

seen in Micro‐

Macro

_duality

_{[1, 2]}

as amathematical version of “

Quantum‐Classical

correpsondence”’

between

_microscopic

intra‐sectorfial &

_macroscopic

inter‐

2.2 Inter‐sectorial relations

&

_{Symmetry Breaking}

3)

Mutual relations among different sectors:

disjoint

w.r.\mathrm{t}. unbroken

symmetry

Different sectors are connected

by

theactions of broken

symmetries

: as

explained later,

this contrast is shared even

by

D(H)R

theory

of

unbroken

_symmetry!

4)

Emergence

process

[

Macro

\Leftarrow \mathrm{M}\mathrm{i}\mathrm{c}\mathrm{r}\mathrm{o}|

of

Spec

=\mathrm{s}\mathrm{e}\mathrm{c}\mathrm{t}\mathrm{o}\mathrm{r}

‐classifying

spacevia

forcing along

(generic)

filters

This is controlled

_{mathematically}

_by

Tomita theorem to

_decompose

a

Hilbert bimodule

$\pi$(\mathcal{X})^{\prime\prime\overline{\mathcal{X}}}L^{\infty}(E_{\mathcal{X}})

_{:= $\pi$(\mathcal{X})^{u}\otimes L^{\infty}(E_{\mathcal{X}})}

withleft

_{$\pi$(\mathcal{X})}

&

_right

L^{\infty}(E_{\mathcal{X}}, $\mu$)

actions,

via centralmeasure _$\mu$

supported by

Spec=\mathrm{s}\mathrm{u}\mathrm{p}\mathrm{p}( $\mu$)=

Sp(3)\subset F_{\mathcal{X}}

: factorstates

(

\subset E_{\mathcal{X}}

: state _{space of \mathcal{X}}

).

\Rightarrow

_Applications

tostatistical inference basedon

large

deviation

princi‐

ple

[3]

andto derivation of Born rule

_[4].

2.3

_Simplex

vs.

complex/

short vs.

long

exact sequences

5)

In

_homological

_algebra,

distinctions between individual modules and

complexes

of modules and between short and

_long

exact _sequences

are well known tobe

important.

For \mathrm{a}

(

\infty

‐dimensional)

vonNeumann

algebra

\mathcal{X}_,

algebra

\mathcal{X} and itscom‐

mutant \mathcal{X} are

symmetric

in standard

form

viaTomita‐Takesaki modular

conjugation

J:\mathcal{X}\ni x\ovalbox{\tt\small REJECT} JxJ\in \mathcal{X}

\Rightarrow \mathrm{v}\mathrm{i}\mathrm{a} the

_degrees

of freedom of thecommutant.\mathcal{X}

_‘,

a

complex

(X_{n})_{n\in \mathrm{N}}

of \mathcal{X}‐modules can

easily

be reducedto an \mathcal{X}-module

\displaystyle \bigoplus_{n}X_{n}

:

X:=\displaystyle \bigoplus_{n}X_{n}

\{X_{n}=Xp_{n}

with

_{p_{n}\in Proj(\mathcal{X})\}_{n\in \mathrm{N}}.}

Thus, complexes

of

modules become redundant in the

_category

_{Mod_{\mathcal{X}}}

of \mathcal{X}‐modules with \infty-\mathrm{d}\mathrm{i}\mathrm{m}' \mathrm{a}\mathrm{l} v.N.

_alg.

\mathcal{X}, where the essence of

long

exact

is reduced to the

_{triangulated category}

_{X\rightarrow Y\rightarrow Z\rightarrow \mathcal{I}(X)}

of \mathcal{X}-modules.

2.4 Ward

Takahashi

identities

&

exact sequences in

_QFT

Thus thedistinctions between individual modules and

_complexes

of mod‐ ules and between short and

_long

exact _sequences are less

important

for

the sake of

_classifying

_{representations}

of the

_algebra

\mathcal{X} of

_physical

vari‐ ables. Suchdistinctions_are,

_however,

still

_meaningful

in_{relation with group‐} theoreticalor

geometric

_aspects

arising

fromthe actions of

dynamics

and/or

symmetries

on the

physical

_systems

in the

following

sense:

Short exact_sequence:

_corresponds

to Ward‐ Takahashiidentities for correlation functions to describe unbroken

_symmetry

Long

exact sequence:

corresponds

to Ward‐Takahashi identities

describing

spontaneo\mathrm{r}\underline{ $\iota$}sly

broken

_symmetries

with Goldstone bosons

tofunction as

connecting morphisms

Inthis

_context,

thecontrast between short vs. exact _{sequences is related}

with unbroken vs. broken

symmetries

and also with the absenceor _presence

of

_connecting

_morphisms.

This last item is

_directly

related withthe fate of Goldstone bosons

_(at

_least,

for

_spontaneous

breakdown of

_symmetry).

2.5 _{Relation between emergence}

&

eventualization

Mutual relation _{between emergence} & eventualization

_(the

latter

_empha‐

sized

_by

Dr.

_Saigo):

while the former refers to universal transitions

_(real

or

virtual)

from States to

_Spec

asthe level of

classifying

_{spaces within}

discussed

_contexts,

the latter

_concept,

_{eventualization,}

means the actual

physical

processes,

typically taking place

in

experimental situations,

which

verify

the relevance and

_actuality

of the

_points

_belonging

to

_Spec

asthe real‐

ized form ofevents. This context is described

_by

the

_expression,

events \in

Spec

mainly

materialized in the

_{quantum‐mechanical}

_{measurement pro‐}

cesses. In contrast to this

_{quantum‐mechanical}

_context,

“localization. of

fields” describes transitions from

_quantum

fieldsto classical fields.

2.6

_{Symmetry Breaking}

&

_{Classifying Space}

6)

Symmetry Breaking

EY

_Emerg

ence

of Classifying Space

Sector‐classifying

space emerges

typically

from

spontaneous

breakdown of

_symmetry

of a

dynamical

_system

X\cap G with action of a _group G

(“spontaneous”

=\mathrm{n}\mathrm{o}

changes

in

dynamics

ofthe

system).

Criterion

_for

_{Symmetry Breaking}

_([1]

SB

_criterion,

for

_short):

_judged

by non‐triviality

of central

_dynamical

_system

_{3_{ $\pi$}(\mathcal{X})\cap G}

_arising

from the

original

one \mathcal{X}\cap G

I.e.,

symmetry

G is broken in sectors

_{\in Sp(3)}

with non‐trivial re‐

sponses to central G‐action.

The G

_{‐transitivity assumption}

with unbroken

_subgroup

H in broken G leads to such a

specific

form of

sector‐classifying

_space as

G/H.

\Rightarrow Classical

_geometric

structureon

G/H

arises

physically

fromemer‐

gence process via condensation ofa

family

of

degenerate

_{vacua, each}

of.which is

_{mutually distinguished}

_by

condensed values

_{\in Sp(3)=G/H.}

2.7 Sector Bundle

&

_Logical

Extension from. const to vari‐ able

In _{this way,} \infty‐number of

low‐energy

quanta

are condensed into

_geometry

of classical Macro

_objects

_{\in G/H.}

In combinationwith sector structure

\hat{H}

ofunbroken

_symmetry

H

_(à

la

DHR‐DR

_theory),

total sector structure due to this

_symmetry

_breaking

is described

_by

a sector bundle

G\times\hat{H}

with fiber

\hat{H}

over base _space

G/H

consisting

of

_“degenerate

vacu

a^{H}[l

, 5

].

When this

_geometric

structureis

_established,

all the

_physical

_quantities

are

parametrized by

condensed values

of

order

parameters

\in G/H

\Rightarrow

_Logical

extension”’ ofconstants

₍

=

global

objects)

into sector‐

dependent function objects

(:

origin

of local _gauge

_structures)

2.8

_Symmetric

_Space

Structure of

_G/H

This

_homogeneous

space

G/H

is a

symmetric

_{space with} Cartan involu‐

tion

_(as

shown

_{here) [IO,}

in

_{preparation].}

’

Lie‐bracket relations

_{[\mathfrak{h}, \mathfrak{h}]\subset \mathfrak{h}, [\mathfrak{h}, \mathrm{m}]\subset \mathfrak{m}}

hold for Lie structures_{\mathfrak{g},}

_{\mathfrak{h},}

\mathrm{m}

ofGHM

_:=G/H.

If

_{[\mathfrak{m}\mathfrak{m}]\subset \mathfrak{h}}

is

_verified,

M becomesa

symmetric

_space

(at

least,

locally)

equipped

withCartan involution \mathcal{I} with

_eigenvalues

_{\mathcal{I}\mathrm{r}_{\mathfrak{h}}=+1}

&

_{\mathcal{I}\mathrm{f}_{\mathrm{m}}=-1}

:

Proof of

_{[\mathfrak{m},\mathrm{m}]\subset \mathfrak{h})}

_{[\mathfrak{m},\mathrm{m}]=}

_holonomy

associatedwith aninfinitesimal

loop

in inter‐sectorial _space

_M=Sp(3)

_along

broken direction

\Rightarrow[\mathrm{m}, \mathrm{m}]=

effect of broken G transformation

_along

an infinitesimal

loop

6

on M

starting

from and

returning

to the same

$\gamma$\in M.

\Rightarrow $\iota$

\mathfrak{n}-component

in

_{[\mathfrak{m}, \mathfrak{m}]}

is absent

_by

the aboveSBcriterion.

_Thus,

_M=G/H=

Sp(3)

is a

symmetric

_space

(at

least,

locally).

2.9

_Example

1: Lorentz boosts

Typical

example

of this sort can be found for Lorentz _group

\mathcal{L}_{+}^{\uparrow}=:G,

rotation group

SO(3)=:H,

G/H=M\cong \mathbb{R}^{3}

:

symmetric

_space of Lorentz

For

_\mathfrak{h}

_{:=\{M_{ij};i,j=1, 2, 3, i<j\},}

\mathfrak{m}

:=\{M_{0i};i=1, 23\}

, the relations

[\mathfrak{h}, \mathfrak{h}]=\mathfrak{h}, [\mathfrak{h}, \mathfrak{m}]=\mathfrak{m}

,

[m,m]

\subset \mathfrak{h}

followfromthe basic Lie

algebra

structure:

[iM_{ $\mu \nu$}, iM_{$\rho$_{ $\sigma$}}]=-($\eta$_{\mathrm{v} $\rho$}iM_{ $\mu \sigma$}-$\eta$_{v $\sigma$}iM_{ $\mu \rho$}-$\eta$_{ $\mu \rho$}iM_{ $\nu \sigma$}+$\eta$_{ $\mu \sigma$}iM_{ $\nu \rho$})

.

In contrast to the usual

_{interpretation}

of unbroken

_\mathfrak{h}

& \mathfrak{m} unbroken

Lorentz boosts \mathrm{m} is

speciality of

the vacuum

situation,

which is due

to such results as Borchers‐Arveson theorem

(:

Poincaré

_generators

can be

physical

observables

_only

in vacuum

representation)

& as the

_spontaneous

breakdown ofLorentz boosts at

_{T\neq 0K[6].}

Thus Lorentzframes

M\cong \mathbb{R}^{3}

with

_{[boost, boost]}

=

rotation, give

a

typi‐

cal

_example

of

_symmetric

_spacestructure

_emerging

from

_symmetry

_breaking.

2.10

_Example

2: 2nd Law of

_{Thermodynamics}

Along

this

_line,

chiral

_symmetry

with current

_algebra

structure

_{[V, V]=}

V,

[V, A]=A, [A, A]=V

and

_conformal

_symmetry

also

_provide

_typical

examples.

Physically

more

interesting example

canbefound in

thermodynamics:

1st law of

_{thermodynamics \Rightarrow\triangle Q\rightarrow $\Delta$ E=\triangle Q+ $\Delta$ W\rightarrow $\Delta$ W}

:

exact sequence

corresponding

to

_{\mathfrak{h}\rightarrow \mathfrak{g}\rightarrow \mathrm{m}=\mathfrak{g}/\mathfrak{h}.}

With

_respect

to Cartan involution with +

_assigned

to heat

_production

\triangle Q

and—to

_macroscopic

work $\Delta$ W, the

holonomy

[\mathfrak{m}, \mathfrak{m}]\subset \mathfrak{h}

correspond‐

ing

to a

loop

in the space M of

thermodynamic

variables becomes

just

Kelvins version

_of

2nd law

_of

_{thermodynamics}

namely, holonomy

[\mathfrak{m}\mathfrak{m}]

in the

_cyclic

process with

$\Delta$ E= $\Delta$ Q+\triangle W=0,

describes heat

_production

_{\triangle Q\geq 0:- $\Delta$ W=-[\mathrm{m}, \mathfrak{m}]= $\Delta$ Q>0}

_(from

system

to

_outside)

2.11

Sector Bundle&

_Holonomy

In use ofsector bundle

\hat{H}\rightarrow G_{H}\times\hat{H}\rightarrow G/H

,

physical

origin

of

space‐time

concept

can be seenin its

physical

_{emergence process}

[7].

For

_simplicity,

we assume here that a _group G ofbroken internal_sym‐

metry

be extended

_by

a _group \mathcal{R} of

space‐time

_symmetry

(typically

trans‐

lations)

into a

larger

_group $\Gamma$=\mathcal{R}\times G defined

by

asemi‐direct

product

of

\mathcal{R} & G with

_{$\Gamma$/G=\mathcal{R}.}

In this _case, thesector bundles have a double fibrationstructure:

\hat{H} \rightarrow G_{H}\times\hat{H} \rightarrow $\Gamma$_{GHH}\times(G\times\hat{H})= $\Gamma$\times\hat{H}

\downarrow

\downarrow

2.12

_Holonomy

_along

Goldstone condensates

\RightarrowThree differentaxesondifferent levels in

\mathrm{S}\mathrm{p}\mathrm{e}\mathrm{c}=\mathrm{s}\mathrm{e}\mathrm{c}\mathrm{t}\mathrm{o}\mathrm{r}‐classifying

_space:

a)

sectors

\hat{H}

of unbroken

_symmetry

_H,

b)

\mathrm{d}\mathrm{e}\mathrm{g}

. vacua

G/H=M

due to broken internal

symmetry

[1, 5],

c) $\Gamma$/G=\mathcal{R}

as

_emergent

_space‐time

[7]

in broken external

_symmetry.

These axes arise in aseries of

_{structure‐group}

contractions H\leftarrow G\leftarrow $\Gamma$

of

_principal

bdles

_{P_{H}\llcorner\rightarrow P_{G}\rightarrow P_{ $\Gamma$}}

over \mathcal{R},

specified

by solderings

as bdle

sections,

\mathcal{R}\rightarrow $\rho$ P_{G}/H=P_{H_{H}^{\times}}(G/H)

,

\mathcal{R}\rightarrow $\tau$ P_{ $\Gamma$}/G=P_{G_{G}^{\times}}( $\Gamma$/G)

=P_{G_{G}^{\times}}\mathcal{R}

,

corresponding physically

to Goldstone modes:

P_{H}

\llcorner\rightarrow

P_{G}

\rightarrow

P_{ $\Gamma$}

H\downarrow \mathcal{R}

\rightarrow

_{P_{G}/H\downarrow H}

\rightarrow

_{P_{ $\Gamma$}/H\downarrow H}

\backslash \backslash O $\iota$_{\mathcal{R}}G/H \rightarrow \mathrm{O} $\tau$ P_{ $\Gamma$}/G\downarrow G/H

\backslash \backslash O

\downarrow \mathcal{R}

\mathcal{R}

2.13

_Helgason

_duality

with Hecke

_algebra

From

_{algebraic viewpoint}

_(which

isdualto the

_Helgason

_duality

_{K\backslash G\leftrightarrow}

\nearrow K\backslash G/H \nwarrow

G/H

:

K\backslash G

\leftrightarrow

G/H

with Radon transforms & Hecke

alge‐

\nwarrow G \nearrow

bra

_{K\backslash G/H)}

, the essence of the relevant structures can be viewed as the

“stereo‐graphic”

extension of such

_{planar diagrams}

as

controlling

“aug‐

mented

_algebras”’

_[1]

of crossed

_products

to describe

_symmetry

_breaking:

\displaystyle \frac{G}{\mathcal{X}}H/H\swarrow \mathcal{X}^{H}=\overline{\mathcal{X}}^{G}\searrow H\mathcal{X}

_{\mathcal{R}\swarrow \mathcal{O}_{ $\rho$}=\mathcal{O}_{d}^{H}\Downarrow\searrow H\mathcal{O}_{d}}

[same

sort

\displaystyle \downarrow H\searrow\searrow\swarrow G/H\frac{\Downarrow}{\mathcal{X}}

\downarrow

\Leftrightarrow A(\mathcal{R})

$\iota$_{H}\searrow\searrow \mathcal{X}(\mathcal{R})\swarrow \mathcal{R}\downarrow

: of lines are

in the same

\displaystyle \frac{\downarrow}{H\backslash G}\swarrow\Downarrow $\zeta$\rightarrow\hat{G}\searrow\searrow\rightarrow

\hat{H}\downarrow

\hat{\mathcal{R}}\downarrow\sim\swarrow*

\hat{ $\Gamma$}\Downarrow

\searrow\searrow\rightarrow\hat{H}\downarrow

exact

_seq]

Note that

_push‐out

_diagram

inDRreconstructionof field

_algebra

_{\mathcal{X}(\mathcal{R})}

shows_up here

_(right)

in

_spite

of its unbroken

_symmetry.

3

_Symmetric

_{space structure}

&

_{Maxwell‐type}

_equa‐

tions

Symmetric

space structures of

G/H=M

&

$\Gamma$/G=\mathcal{R}

due to

_symmetry

breakingo equation

of

_type

_{[\mathfrak{m},\mathfrak{m}]\subseteq \mathfrak{h}}

,whichconnects

holonomy

[\mathfrak{m}, \mathfrak{m}] (in

Notethat this feature is shared incommon

_by

Maxwell&_{Einstein equa‐}

tions of

_{electromagnetism}

and of

_gravity,

_{respectively:}

LHS:

_{(curvature F_{ $\mu$ \mathrm{v}}}

or

R_{ $\mu \nu$})

=

(source

current

J_{ $\mu$}

_or

T_{ $\mu$ v})

: RHS.

According

to 2nd Noether theorem

_(developed

in the

_theory

of invari‐

ants),

Maxwell

_equation

isan

identity following

from the invariance of action

integral

under

_space‐time

_dependent

transformations.

In

contrast,

nosuch classical

_quantities

asaction

integrals

nor

Lagrangian

densitiesareavailable inour

algebraic

&

categorical

formulation of

_quantum

fields.

3.1

Galois

Functor in

_{Doplicher‐Roberts}

reconstruction of

symmetry

The

_expected

roles ofaction

_integral:

to determine

_{representation}

contents

of a

theory

\Rightarrow \mathrm{c}\mathrm{a}\mathrm{n} be substituted

_{by categorical}

data

_concerning

Galois

group due to

_Doplicher

& Roberts

_(DR),

in terms of DR

_category

\mathcal{I} of

modules of local excitations:

Obj(T)

: local

endomorphisms

$\rho$\in End(A)

of observable

alg.

A_,selected

by

DHRlocalization criterion

_{$\pi$_{0}0 $\rho$(_{A(O')}\cong $\pi$ 0\mathrm{r}_{A(\mathrm{O}')},}

\mathrm{M}\mathrm{o}\mathrm{r}(T):T\in T( $\rho$\leftarrow $\sigma$)\subset A

intertwining

$\rho$,

$\sigma$\in\cdot T: $\rho$(A)T=\wedge T $\sigma$(A)

.

The group H of unbroken internal

_symmetry

arises as the group H=

End_{\otimes}(V)

of

_unitary

tensorial

₍

=

monoidal)

natural transformations_{u :} V\leftarrow

V with the

_{representation}

functor V:T\leftrightarrow Hilb to embed T into

_category

Hilb of Hilbert spaces with

_morphisms

as bounded linear maps.

3.2 Galois FUnctor in

_Category

&

_{its Local gauge}

invariance

V( $\rho$) \leftarrow v_{ $\rho$} W(p)

In viewof

_{commutativity}

_diagrams:

_V(T)\uparrow

O

_{\uparrow W(T)\mathrm{i}.\mathrm{e}.v_{ $\rho$}W(T)=}

V( $\sigma$) \leftarrow v_{ $\sigma$} W( $\sigma$)

V(T)v_{ $\sigma$}

with

_{T\in T(p\leftarrow $\sigma$)}

, in the definition of natural transformation

v:V\leftarrow W, we

try

here to

reinterpret

it as a

categorical

definition ofa local

gauge

transformation

W\rightarrow$\tau$_{v}$\tau$_{v}(W)=V

ofafunctor W into V on the basis

of definition:

$\tau$_{v}(W)(T):=v_{ $\rho$}W(T)v_{ $\sigma$}^{-1}

for

_{T\in T( $\rho$\leftarrow $\sigma$)}

.

Similar formula canbe found for gauge links in lattice gauge

_theory.

Then,

the

_{commutativity,}

_{u_{ $\rho$}V(T)=V(T)u_{ $\sigma$}}

for

_{u\in End_{\otimes}(V)}

, can be

interpreted

as local _{gauge invariance}

$\tau$_{\mathrm{u}}(V)=V

of the functor V under

local_gauge

_{transformation}

_{V\rightarrow$\tau$_{u}(V)}

induced

_by

anaturaltransformation

3.3 _{Local gauge invariance & Maxwell}

_equation

Inthe

_{original Doplicher‐Roberts theory,}

local

_{endomorphisms}

$\rho$\in T\subset End(\mathcal{A})

have, unfortunately,

been

_{regarded global}

constant

objects, owing

to the

_emphasis

on

space‐time

transportabilityl,

and

hence,

the

_left‐right

difference of _{u_{ $\rho$}} and u_{ $\sigma$} in

$\tau$_{\mathrm{u}}(V)(T):=u_{ $\rho$}V(T)u_{ $\sigma$}^{-1}

has not

been

_properly

_recognized

as

important

signal

of local gauge structures.

From the

_{general viewpoint}

of

_{forcing method, however,}

the essential

features of

_logical

extension

_from

constants tovariables

_naturally

leadto

the

_{interpretation}

of

_{$\tau$_{u}(V)(T)=u_{ $\rho$}V(T)u_{ $\sigma$}^{-1}=V(T)}

asthe characterization

of local gauge invariance of V under local gauge transform

u:\mathcal{I}\ni p\mapsto u_{ $\rho$}.

This is in

_harmony

also with the alternative formulation of

_principal

bundles interms of

_{group‐valued Čech}

_{cohomologies.}

3.4

_{Symmetry breaking}

&

Maxwell

_equation

In the above

_{preliminary discussion,}

the recovered _{group H of unbroken}

symmetry

is

_compact

in DR

_{theory. So,}

the _space H ofsector

_parameters

is

_discrete,

which makes it difficult to

_incorporate

differential

_equationsr

To

_adapt

the rolesof DR

_category

_{\mathcal{I}\subset End(\mathcal{A})=End(\mathcal{X}^{H})}

indetermin‐

ing

the factor

_spectrum

=H=Sp(3(\mathcal{X}^{H}))=\hat{H}

to our

present

_purpose, weneedto

replace

\mathcal{I}

_by

T==End(\mathcal{X})

with

\mathcal{X}=\mathcal{X}^{H}\rangle\triangleleft\hat{\mathcal{R}}

and with

_{$\Gamma$/G=\mathcal{R}(}

: _space‐

time)

in the

_two‐step

construction of

_{augmented algebras}

associated with

the series of group extensions: unbroken H\mathrm{c}\rightarrow broken internal G\mathrm{c}\rightarrowbroken

external $\Gamma$.

By

repeating

the

_categorical

formulation of

End_{\otimes}(V : T\rightarrow Hilb)

with \mathcal{I} and V

_{replaced by}

T=

and V we can

reproduce

theessenceof 2nd Noether

theoremto connect _{the local gauge invariance and Maxwell}

_equation.

3.5 Second Noether theorem

In this

_context,

2nd Noether theorem can be

generalized

into aform with

three

_type

_{arguments, x\in \mathcal{R},}

_{$\xi$\in G/H, a\in\hat{H}}

, so as to

incorporate

low‐

energy theorem

(with

“soft

pions”)

due to

symmetry

breaking.

For

_simplicity,

we

_repeat

its standardform with infinitesimal local gauge

transformation

_{$\delta$_{ $\Lambda$}$\varphi$^{a}(x)=G^{a}(x)\cdot $\Lambda$(x)+T^{a $\mu$}(x)\cdot\partial_{ $\mu$} $\Lambda$(x)}

offields

_{$\varphi$^{a}(x)}

_spec‐ ified

_by

an “inifinitesimal

parameter”’

$\Lambda$= $\Lambda$(x)

ofanatural transformation

depending

on sector

parameter

x\in \mathcal{R}.

Then

_{Maxwell‐type}

_equation

holds

_identically,

\partial_{ $\nu$}K^{ $\nu \mu$}+J^{ $\mu$}=0,

when K^{ $\nu \mu$} and J^{ $\mu$} are “defined” in relation with the (‘infinitesimal trans‐

forms”’ of Galois functor V:

K^{ $\nu \mu$} :

=T^{a $\mu$}\displaystyle \frac{\partial}{\partial(\partial_{ $\nu$}$\varphi$^{a})}V,

J^{ $\mu$} :

=T^{a $\mu$}(\displaystyle \frac{\partial}{\partial$\varphi$^{a}}-\frac{\partial}{\partial(\partial_{ $\nu$}$\varphi$^{a})})V+G^{a}\frac{\partial}{\partial(\partial_{ $\mu$}$\varphi$^{a})}V.

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