Masujima M. Applied Mathematical Methods in Theoretical Physics

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418 10 Calculus of Variations: Applications

The conformal charge K

generates the conformal transformation as it should.

In contradistinction to the inﬁnitesimal transformation forms discussed above,

the geometrical meaning of the scale transformation and the conformal transfor-

mation becomes clear in the ﬁnite transformation forms

(

scale transformation x

µ

= exp[ρ]x

conformal transformation x

µ

= (x

− c

)(1 −2cx + c

)

−1

The conformal group as a whole is not the invariance group of the nature. The

mass term in the Lagrangian density violates the scale invariance. If the scale

transformation is the symmetry transformation of the theory, the theory must

possess the continuous mass spectrum, which is not the case in the nature. Still

for the conformal ﬁeld theory, the conformal group is important.

All of the transformations discussed above, speciﬁcally, the space–time trans-

lation, the Lorentz transformation , the scale transformation and the conformal

transformation are space–time coordinate dependent. Generally, for the space–time

coordinate transformation,wehaveδ

= 0. On the other hand, for the internal

transformation,wehaveδ

= 0.

The global internal symmetry, which is space–time independent, results in the

internal current conservation. Extension to the local internal symmetry which is

space–time dependent is accomplished with the introduction of the appropriate

gauge ﬁeldsby invoking Weyl’s gauge principle. This has been discussed in the

next section.

10.9

Weyl’s Gauge Principle

In electrodynamics, we have a property known as the gauge invariance.Inthetheory

of gravitational ﬁeld, we have a property known as the scale invariance.Beforethe

birth of quantum mechanics, H. Weyl attempted to construct the uniﬁed theory

of classical electrodynamics and gravitational ﬁeld. But he failed to accomplish his

goal. After the birth of quantum mechanics, he realized that the gauge invariance

of electrodynamics is not related to the scale invariance of the gravitational ﬁeld,

but is related to the invariance of the matter ﬁeld φ(x) under the local phase

transformation. The matter ﬁeld in interaction with the electromagnetic ﬁeld has

a property known as the charge conservation law or the current conservation law .In

this section, we discuss Weyl’s gauge principle for the U(1) gauge ﬁeld and the

non-Abelian gauge ﬁeld, and Kibble’s gauge principle for the gravitational ﬁeld.

Weyl’s gauge principle: Electrodynamics is described by the total Lagrangian

density with the use of the four-vector potential A

(x)by

tot

= L

matter

(φ(x), ∂

φ(x)) +L

int

(φ(x), A

(x)) + L

gauge

(x), ∂

(x)).

(10.9.1)

10.9 Weyl’s Gauge Principle 419

This system is invariant under the local U(1) transformations

(x) → A



(x) ≡ A

(x) − ∂

ε(x), (10.9.2)

φ(x) → φ



(x) ≡ exp[iqε(x)]φ(x). (10.9.3)

The interaction Lagrangian density L

int

(φ(x), A

(x)) is generated by the substitution

∂

φ(x) → D

φ(x) ≡ (∂

+ iqA

(x))φ(x), (10.9.4)

in the original matter ﬁeld Lagrangian density L

matter

(φ(x), ∂

φ(x)). The derivative

φ(x) is called the covariant derivative of φ(x) and transforms exactly like φ(x),

φ(x) → (D

φ(x))



= exp[iqε(x)]D

φ(x), (10.9.5)

under the local U(1) transformations, Eqs. (10.9.2) and (10.9.3).

The physical meaning of this local U(1) invariance lies in its weaker version, the

global U(1) invariance, namely,

ε(x) = ε, space–time independent constant.

The global U(1) invariance of the matter ﬁeld Lagrangian density,

matter

(φ(x), ∂

φ(x)),

under the global U(1) transformation of φ(x),

φ(x) → φ



(x) = exp[iqε]φ(x), ε = constant, (10.9.6)

in its inﬁnitesimal version,

δφ(x) = iqεφ(x), ε = inﬁnitesimal constant, (10.9.7)

results in

∂L

matter

(φ(x), ∂

φ(x))

∂φ(x)

δφ(x) +

∂L

matter

(φ(x), ∂

φ(x))

∂(∂

φ(x))

δ(∂

φ(x)) = 0. (10.9.8)

With the use of the Euler–Lagrange equation of motion for φ(x),

∂L

matter

(φ(x), ∂

φ(x))

∂φ(x)

− ∂

∂L

matter

(φ(x), ∂

φ(x))

∂(∂

φ(x))

= 0,

we obtain the current conservation law

∂

matter

(x) = 0, (10.9.9)

420 10 Calculus of Variations: Applications

εJ

matter

(x) =

∂L

matter

(φ(x), ∂

φ(x))

∂(∂

φ(x))

δφ(x). (10.9.10)

This in its integrated form becomes the charge conservation law,

matter

(t) = 0, (10.9.11)

matter

(t) =





matter

(t,



x ). (10.9.12)

Weyl’s gauge principle considers the analysis backward. The extension of the

‘‘current conserving’’ global U(1) invariance, Eqs. (10.9.7) and (10.9.8), of the

matter ﬁeld Lagrangian density L

matter

(φ(x), ∂

φ(x)) to the local U(1) invariance

necessitates

(1) the introduction of the U(1) gauge ﬁeld A

(x), and the replacement of the

derivative ∂

φ(x) in the matter ﬁeld Lagrangian density with the covariant derivative

φ(x),

∂

φ(x) → D

φ(x) ≡ (∂

+iqA

(x))φ(x), (10.9.4)

and

(2) the requirement that the covariant derivative D

φ(x) transforms exactly like the

matter ﬁeld φ(x)underthelocalU(1) phase transformation of φ(x),

φ(x) → (D

φ(x))



= exp[iqε(x)]D

φ(x), (10.9.5)

under

φ(x) → φ



(x) ≡ exp[iqε(x)]φ(x). (10.9.3)

From requirement (2), we obtain the transformation law of the U(1) gauge ﬁeld

(x) immediately,

(x) → A



(x) ≡ A

(x) − ∂

ε(x). (10.9.2)

From requirement (2), the local U(1) invariance of L

matter

(φ(x), D

φ(x)) is also

self-evident. In order to give dynamical content to the U(1) gauge ﬁeld, we

introduce the ﬁeld strength tensor F

µν

(x) to the gauge ﬁeld Lagrangian density

gauge

(x), ∂

(x)) by the trick,

, D

]φ(x) = iq(∂

(x) −∂

(x))φ(x) ≡ iqF

µν

(x)φ(x). (10.9.13)

10.9 Weyl’s Gauge Principle 421

From the transformation law of the U(1) gauge ﬁeld A

(x), Eq. (10.9.2), we observe

that the ﬁeld strength tensor F

µν

(x) is the locally invariant quantity

µν

(x) → F



µν

(x) = ∂



(x) − ∂



(x) = ∂

(x) − ∂

(x) = F

µν

(x).

(10.9.14)

As the gauge ﬁeld Lagrangian density L

gauge

(x), ∂

(x)), we choose

gauge

(x), ∂

(x)) =−

µν

(x)F

µν

(x). (10.9.15)

In this manner, we obtain the total Lagrangian density of the matter-gauge system

which is locally U(1) invariant as

tot

= L

matter

(φ(x), D

φ(x)) +L

gauge

µν

(x)). (10.9.16)

We obtain the interaction Lagrangian density L

int

(φ(x), A

(x)) as

int

(φ(x), A

(x)) = L

matter

(φ(x), D

φ(x)) −L

matter

(φ(x), ∂

φ(x)), (10.9.17)

which is the universal coupling generated by Weyl’s gauge principle. As a result of

the local extension of the global U(1) invariance, we derived the electrodynamics

from the current conservation law, Eq. (10.9.9), or the charge conservation law,

Eq. (10.9.11).

We shall now consider the extension of the present discussion to the non-Abelian

gauge ﬁeld. We let the semisimple Lie group G be the gauge group. We let the

representation of G in the Hilbert space be U(g), and its matrix representation on

the ﬁeld operator

(x) in the internal space be D(g),

U(g)

(x)U

−1

(g) = D

n,m

(g)

(x), g ∈ G. (10.9.18)

For the element g

∈ G continuously connected to the identity of G by the parameter

{ε

}

α=1

,wehave

U(g

) = exp[iε

] = 1 +iε

+···, T

: generator of Lie group G,

(10.9.19)

D(g

) = exp[iε

] = 1 +iε

+···, t

: realization of T

(x),

(10.9.20)

, T

] = iC

αβγ

, (10.9.21)

, t

] = iC

αβγ

. (10.9.22)

422 10 Calculus of Variations: Applications

We shall assume that the action functional I

matter

[ψ

] of the matter ﬁeld

Lagrangian density L

matter

(ψ

(x), ∂

(x)), given by

matter

[ψ

] ≡



x L

matter

(ψ

(x), ∂

(x)), (10.9.23)

is invariant under the global G transformation,

δψ

(x) = iε

)

n,m

(x), ε

= inﬁnitesimal constant. (10.9.24)

Namely, we have

∂L

matter

(ψ

(x), ∂

(x))

∂ψ

(x)

δψ

(x) +

∂L

matter

(ψ

(x), ∂

(x))

∂(∂

(x))

δ(∂

(x)) = 0.

(10.9.25)

With the use of the Euler–Lagrange equation of motion,

∂L

matter

(ψ(x ), ∂

ψ(x))

∂ψ

(x)

− ∂

(

∂L

matter

(ψ(x ), ∂

ψ(x)

∂(∂

(x))

) = 0, (10.9.26)

we have the current conservation law and the charge conservation law,

∂

α,matter

(x) = 0, α = 1, ..., N, (10.9.27a)

where the conserved matter current J

α,matter

(x)isgivenby

α,matter

(x) =

∂L

matter

(ψ(x ), ∂

ψ(x))

∂(∂

(x))

δψ

(x), (10.9.27b)

and

matter

(t) = 0, α = 1, ..., N, (10.9.28a)

where the conserved matter charge Q

matter

(t)isgivenby

matter

(t) =





α,matter

(t,



x ), α = 1, ..., N. (10.9.28b)

Invoking Weyl’s gauge principle, we extend the global G invariance of the matter

system to the local G invariance of the matter-gauge system under the local G

phase transformation,

δψ

(x) = iε

(x)(t

)

n,m

(x). (10.9.29)

Weyl’s gauge principle requires the following:

10.9 Weyl’s Gauge Principle 423

(1) the introduction of the non-Abelian gauge ﬁeld A

αµ

(x) and the replacement

of the derivative ∂

(x) in the matter ﬁeld Lagrangian density with the covariant

derivative (D

ψ(x))

∂

(x) → (D

ψ(x))

≡ (∂

n,m

+ i(t

)

n,m

γµ

(x))ψ

(x), (10.9.30)

and

(2) the requirement that the covariant derivative (D

ψ(x))

transforms exactly like

the matter ﬁeld ψ

(x)underthelocalG phase transformation of ψ

(x), Eq. (10.9.29),

δ(D

ψ(x))

= iε

(x)(t

)

n,m

ψ(x))

, (10.9.31)

where t

is the realization of the generator T

upon the multiplet ψ

(x).

From Eqs. (10.9.29) and (10.9.31), the inﬁnitesimal transformation law of the

non-Abelian gauge ﬁeld A

αµ

(x) follows,

δA

αµ

(x) =−∂

(x) + iε

(x)(t

adj

)

αγ

γµ

(x) (10.9.32a)

=−∂

(x) + ε

(x)C

βαγ

γµ

(x). (10.9.32b)

Then the local G invariance of the gauged matter ﬁeld Lagrangian density

matter

(ψ(x), D

ψ(x)) becomes self-evident as long as the ungauged matter ﬁeld

Lagrangian density L

matter

(ψ(x), ∂

ψ(x)) is globally G invariant.

In order to provide dynamical content to the non-Abelian gauge ﬁeld A

αµ

(x), we

introduce the ﬁeld strength tensor F

γµν

(x) by the following trick:

, D

]ψ(x) ≡ i(t

γµν

(x)ψ (x), (10.9.33)

γµν

(x) = ∂

γν

(x) − ∂

γµ

(x) − C

αβγ

αµ

(x)A

βν

(x). (10.9.34)

We can easily show that the ﬁeld strength tensor F

γµν

(x) undergoes the local G

rotation under the local G transformations, Eqs. (10.9.29) and (10.9.32a), under the

adjoint representation

δF

γµν

(x) = iε

(x)(t

adj

)

γβ

βµν

(x) (10.9.35a)

= ε

(x)C

αγβ

βµν

(x). (10.9.35b)

As the Lagrangian density of the non-Abelian gauge ﬁeld A

αµ

(x), we choose

gauge

γµ

(x), ∂

γµ

(x)) ≡−

γµν

(x)F

µν

(x). (10.9.36)

424 10 Calculus of Variations: Applications

The total Lagrangian density L

total

of the matter-gauge system is given by

total

= L

matter

(ψ(x ), D

ψ(x)) +L

gauge

γµν

(x)). (10.9.37)

The interaction Lagrangian density L

int

consists of two parts due to the nonlinearity

of the ﬁeld strength tensor F

γµν

(x)withrespecttoA

γµ

(x),

int

= L

matter

(ψ(x ), D

ψ(x)) −L

matter

(ψ(x ), ∂

ψ(x))

gauge

γµν

(x)) − L

quad

gauge

γµν

(x)), (10.9.38)

which provides the universal coupling just like the U(1) gauge ﬁeld theory.

The conserved current J

α,total

(x)andtheconservedcharge{Q

total

(t)}

α=1

after the

extension to the local G invariance also consist of two parts,

α,total

(x) ≡ J

µ gauged

α,matter

(x) +J

α,gauge

(x) ≡

δI

total

[ψ, A

αµ

]

δA

αµ

(x)

, (10.9.39a)

total

[ψ, A

αµ

] =



x {L

matter

(ψ(x ), D

ψ(x)) +L

gauge

γµν

(x))}, (10.9.39b)

total

(t) = Q

matter

(t) +Q

gauge

(t) =





x{J

0gauged

α,matter

(t,



x ) + J

α,gauge

(t,



x )}.

(10.9.40)

We note that the gauged matter current J

µ gauged

α,matter

(x) of Eq. (10.9.39a) is not identical

to the ungauged matter current J

α,matter

(x) of Eq. (10.9.27b):

α,matter

(x) of (10.9.27b) =

∂L

matter

(ψ(x ), ∂

ψ(x))

∂(∂

(x))

δψ

(x),

whereas after the local G extension,

µ gauged

α,matter

(x) of (10.9.39a) =

∂L

matter

(ψ(x ), D

ψ(x))

∂(D

(x))

δψ

(x)

∂L

matter

(ψ(x ), D

ψ(x))

∂(D

ψ(x))

iε

)

n,m

(x)

= ε

∂L

matter

(ψ(x ), D

ψ(x))

∂(D

ψ(x))

∂(D

ψ(x))

∂A

αµ

(x)

= ε

∂L

matter

(ψ(x ), D

ψ(x))

∂A

αµ

(x)

= ε

δA

αµ

(x)

gauged

matter

[ψ, D

ψ]. (10.9.41)

Here we note that I

gauged

matter

[ψ, D

ψ] is not identical to the ungauged matter ac-

tion functional I

matter

[ψ

] given by Eq. (10.9.23), but is the gauged matter action

functional deﬁned by

gauged

matter

[ψ, D

ψ] ≡



x L

matter

(ψ(x ), D

ψ(x)). (10.9.42)

10.9 Weyl’s Gauge Principle 425

We emphasize here that the conserved Noether current after the extension of the

global G invariance to the local G invariance is not the gauged matter current

µ gauged

α,matter

(x) but the total current J

α,total

(x), Eq. (10.9.39a). At the same time, we

note that the strict conservation law of the total current J

α,total

(x)isenforcedat

the expense of loss of covariance, which we will see further . The origin of this

problem is the self-interaction of the non-Abelian gauge ﬁeld A

αµ

(x)andthe

nonlinearity of the Euler–Lagrange equation of motion for the non-Abelian gauge

ﬁeld A

αµ

(x).

We shall make a table of the global U(1) transformation law and the global G

transformation law.

Global U(1) transformation law Global G transformation law

δψ

(x) = iεq

(x), charged δψ

(x) = iε

)

n,m

(x), charged

δA

(x) = 0, neutral δA

αµ

(x) = iε

adj

)

αγ

γµ

(x), charged.

(10.9.43)

In the global transformation law of internal symmetry, Eq. (10.9.43), the matter

ﬁelds ψ

(x) which have the group charge undergo global (U(1) or G)rotation.

As for the gauge ﬁelds, A

(x)andA

αµ

(x),theAbeliangaugeﬁeldA

(x)remains

unchanged under global U(1) transformation while the non-Abelian gauge ﬁeld

αµ

(x) undergoes global G rotation under global G transformation. Hence the

Abelian gauge ﬁeld A

(x)isU(1)-neutral while the non-Abelian gauge ﬁeld

αµ

(x)isG-charged. The ﬁeld strength tensors, F

µν

(x)andF

αµν

(x), behave

like A

(x)andA

αµ

(x), under global U(1) and G transformations. The ﬁeld

strength tensor F

µν

(x)isU(1)-neutral, while the ﬁeld strength tensor F

αµν

(x)

is G-charged, which originates from their linearity and nonlinearity in A

(x)and

αµ

(x), respectively.

Global U(1) transformation law Global G transformation law

δF

µν

(x) = 0, neutral δF

αµν

(x) = iε

adj

)

αγ

γµν

(x), charged.

(10.9.44)

When we write the Euler–Lagrange equation of motion for each case,

the linearity and the nonlinearity with respect to the gauge ﬁelds become

clear.

Abelian U(1) gauge ﬁeld non-Abelian G gauge ﬁeld.

∂

νµ

(x) = j

matter

(x), linear D

adj

νµ

(x) = j

α,matter

(x), nonlinear.

(10.9.45)

From the antisymmetry of the ﬁeld strength tensorwith respect to the Lorentz

indices, µ and ν, we have the following current conservation as an identity:

426 10 Calculus of Variations: Applications

Abelian U(1) gauge ﬁeld non-Abelian G gauge ﬁeld

∂

matter

(x) = 0. D

adj

α,matter

(x) = 0.

(10.9.46)

Here we have

adj

= ∂

+ it

adj

γµ

(x). (10.9.47)

As a result of the extension to the local (U(1) or G) invariance, in the case of the

Abelian U(1) gauge ﬁeld, due to the neutrality of A

(x), the matter current j

matter

(x)

alone that originates from the global U(1) invariance is conserved, while in the

case of the non-Abelian G gauge ﬁeld, due to the G-charge of A

αµ

(x), the gauged

matter current j

α,matter

(x) alone that originates from the local G invariance is not

conserved, but the sum with the gauge current j

α,gauge

(x) which originates from the

self-interaction of the non-Abelian gauge ﬁeld A

αµ

(x) is conserved at the expense

of the loss of covariance. A similar situation exists for the charge conservation

law.

Abelian U(1) gauge ﬁeld non-Abelian G gauge ﬁeld

matter

(t) = 0.

tot

(t) = 0.

matter

(t) =



x · j

matter

(t,



x). Q

tot

(t) =



x · j

α,tot

(t,



x ).

(10.9.48)

We discuss the ﬁnite gauge transformation property of the non-Abelian gauge

ﬁeld A

αµ

(x). Under the ﬁnite local G-phase transformation of ψ

(x),

(x) → ψ



(x) = (exp[iε

(x)t

])

n,m

(x), (10.9.49)

we demand that the covariant derivative D

ψ(x) deﬁned by Eq. (10.9.30) transforms

exactly like ψ(x),

ψ(x) → (D

ψ(x))



= (∂

+it



γµ

(x))ψ



(x)

= exp[iε

(x)t

ψ(x). (10.9.50)

From Eq. (10.9.50), we obtain the following equation:

exp[−iε

(x)t

](∂

+ it



γµ

(x)) exp[iε

(x)t

]ψ(x) = (∂

+ it

γµ

(x))ψ (x).

Canceling ∂

ψ(x) term from both sides of the above equation, we obtain

exp[−iε

(x)t

](∂

exp[iε

(x)t

]) +exp[−iε

(x)t

](it



γµ

(x)) exp[iε

(x)t

]

= it

γµ

(x).

10.9 Weyl’s Gauge Principle 427

Solving the above equation for t



γµ

(x), we ﬁnally obtain the ﬁnite gauge transfor-

mation law of A



γµ

(x),



γµ

(x)

= exp[iε

(x)t

]{t

γµ

(x) + exp[−iε

(x)t

](i∂

exp[iε

(x)t

])}exp[−iε

(x)t

(10.9.51)

At ﬁrst sight, we get the impression that the ﬁnite gauge transformation law of



γµ

(x), (10.9.51), may depend on the speciﬁc realization {t

}

γ =1

of the generator

}

γ =1

upon the multiplet ψ (x). Actually, A



γµ

(x) transforms under the adjoint

representation {t

adj

}

γ =1

. The inﬁnitesimal version of the ﬁnite gauge transforma-

tion, (10.9.51), does reduce to the inﬁnitesimal gauge transformation, (10.9.32a)

and (10.9.32b), under the adjoint representation.

Unity of All Forces: Electro-weak uniﬁcation of Glashow–Weinberg–Salam is

based on the gauge group

SU(2)

weak isospin

× U(1)

weak hypercharge

It suffers from the problem of the nonrenormalizability due to the triangular

anomaly in the lepton sector. In the early 1970s, it is discovered that non-Abelian

gauge ﬁeld theory is asymptotically free at short distance, i.e., it behaves like a

free ﬁeld at short distance. Then the relativistic quantum ﬁeld theory of the strong

interaction based on the gauge group SU(3)

color

is invented and is called quantum

chromodynamics.

Standard model with the gauge group

SU(3)

color

× SU(2)

weak isospin

× U(1)

weak hypercharge,

which describes the weal interaction, the electromagnetic interaction and the strong

interaction is free from the triangular anomaly. It suffers, however, from a serious

defect; the existence of the classical instanton solution to the ﬁeld equation in the

Euclidean metric for the SU(2) gauge ﬁeld theory. In the SU(2) gauge ﬁeld theory,

we have the Belavin–Polyakov–Schwartz–Tyupkin instanton solution which is a

classical solution to the ﬁeld equation in the Euclidean metric. A proper account

for the instanton solution requires the addition of the strong CP-violating term to

the QCD Lagrangian density in the path integral formalism. The Peccei–Quinn

axion and the invisible axion scenario resolve this strong CP-violation problem. In

the grand uniﬁed theories, we assume that the subgroup of the grand unifying

gauge group is the gauge group SU(3)

color

×SU(2)

weak isospin

× U(1)

weak hypercharge

We now attempt to unify the weak interaction, the electromagnetic interaction and

the strong interaction by starting from the much larger gauge group G which is

reduced to SU(3)

color

× SU(2)

weak isospin

× U(1)

weak hypercharge

and further down to

SU(3)

color

×U(1)

E.M.