Masujima M. Applied Mathematical Methods in Theoretical Physics

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428 10 Calculus of Variations: Applications

Gravitational ﬁeld: We can extend Weyl’s gauge principle to Utiyama’s gauge

principle and Kibble’s gauge principle to obtain the Lagrangian density for the

gravitational ﬁeld. We note that Weyl’s gauge principle, Utiyama’s gauge principle

and Kibble’s gauge principle belong to the category of the invariant variational

principle.

R. Utiyama derived the theory of the gravitational ﬁeld from his version of the

gauge principle, based on the requirement of the invariance of the action functional

I[φ]underthelocal 6-parameter Lorentz transformation. T.W.B. Kibble derived the

theory of the gravitational ﬁeld from his version of the gauge principle, based

on the requirement of the invariance of the action functional I[φ]underthelocal

10-parameter Poincar´e transformation, extending the treatment of Utiyama.

We let φ represent the set of generic matter ﬁeld variables φ

(x), which we regard

as the elements of a column vector φ(x), and deﬁne the matter action functional

matter

[φ] in terms of the matter Lagrangian density L

matter

(φ, ∂

φ)as

matter

[φ] =



x L

matter

(φ, ∂

φ). (10.9.52)

We ﬁrst discuss the inﬁnitesimal transformationof both the coordinates x

and

the matter ﬁeld variables φ(x),

→ x

µ

= x

+δx

, φ(x) → φ



) = φ(x) +δφ(x), (10.9.53)

where the invariance group G is not speciﬁed. It is convenient to allow the possibility

that the matter Lagrangian density L

matter

explicitly depends on the coordinates x

Then, under the inﬁnitesimal transformation, (10.9.53), we have

δL

matter

≡

∂L

matter

∂φ

δφ +

∂L

matter

∂(∂

φ)

δ(∂

φ) +

∂L

matter

∂x



φ ﬁxed

δx

It is also useful to consider the variation of φ(x)ataﬁxedvalueofx

φ ≡ φ



(x) − φ(x) = δφ − δx

∂

φ. (10.9.54)

It is obvious that δ

commutes with ∂

,sowehave

δ(∂

φ) = ∂

(δφ) −(∂

δx

)∂

φ. (10.9.55)

The matter action functional, (10.9.52), over a space–time region  is trans-

formed under the transformations,(10.9.53), into



matter

[] ≡







matter



)det(∂

µ

x .

10.9 Weyl’s Gauge Principle 429

Thus the matter action functional I

matter

[] over an arbitrary region  is invariant

δL

matter

+ (∂

δx

matter

≡ δ

matter

+ ∂

(δx

matter

) ≡ 0. (10.9.56)

We now consider the speciﬁc case of the Poincar

e transformation,

δx

= iε

+ε

, δφ =

iε

µν

φ, (10.9.57)

where {ε

}and {ε

µν

}with ε

µν

=−ε

νµ

are the 10 inﬁnitesimal constant parameters

of the Poincar

egroup,andS

µν

with S

µν

=−S

νµ

, are the mixing matrices of the

components of a column vector φ(x) satisfying

µν

, S

ρσ

] = i(η

νρ

µσ

+ η

µσ

νρ

− η

νσ

µρ

−η

µρ

νσ

µν

} will be identiﬁed as the spin matrices of matter ﬁeld φ later. From

Eq. (10.9.55), we have

δ(∂

φ) =

iε

ρσ

∂

φ − iε

∂

φ. (10.9.58)

Since we have ∂

(δx

) = ε

= 0, the conditions, (10.9.56), for the invariance of the

matter action functional I

matter

[φ] under the inﬁnitesimal Poincar

e transformations,

(10.9.57), reduce to

δL

matter

≡ 0,

and result in 10 identities,

∂L

matter

∂x

≡ ∂

matter

−

∂L

matter

∂φ

∂

φ −

∂L

matter

∂(∂

φ)

∂

φ ≡ 0, (10.9.59)

∂L

matter

∂φ

ρσ

φ +

∂L

matter

∂(∂

φ)

(iS

ρσ

∂

φ + η

µρ

∂

φ − η

µσ

∂

φ) ≡ 0. (10.9.60)

The conditions (10.9.59) express the translational invariance of the system and are

equivalent to the requirement that L

matter

is explicitly independent of x

.Weuse

the Euler–Lagrange equations of motion in Eqs. (10.9.59) and (10.9.60), obtaining

the ten conservation laws, which we write as

∂

= 0, ∂

ρσ

− x

) = 0, (10.9.61)

≡

∂L

matter

∂(∂

φ)

∂

φ −δ

matter

, S

ρσ

≡−i

∂L

matter

∂(∂

φ)

ρσ

φ. (10.9.62)

430 10 Calculus of Variations: Applications

The ten conservation laws, Eqs. (10.9.59) and (10.9.60), are the conservation laws

of energy, momentum, and angular momentum. Thus {S

µν

} are the spin matrices

of matter ﬁeld φ(x).

We shall also examine the transformations in terms of the variation δ

φ,which

in this case is

φ =−iε

∂

φ +

iε

ρσ

+ x

∂

−x

∂

)φ. (10.9.63)

On comparing with Weyl’s gauge principle, the role of the realizations {t

} of the

generators {T

} upon the multiplet φ is played by the differential operators,

∂

and S

ρσ

+ x

∂

− x

∂

Then, by the deﬁnition of the currents, we expect the currents corresponding to ε

and ε

ρσ

to be given, respectively, by

≡

∂L

matter

∂(∂

φ)

∂

φ and J

ρσ

≡ S

ρσ

− x

. (10.9.64)

In terms of δ

, however, the invariance condition (10.9.56) is not simply

matter

≡ 0, and the additional term δx

∂

matter

results in the appearance of

the term ∂

matter

in the identities (10.9.59) and thus for the term δ

matter

in T

We shall now consider the local 10-parameter Poincar

e transformation in which

the 10 arbitrary inﬁnitesimal constants, {ε

} and {ε

µν

}, in Eq. (10.9.57) become the

10 arbitrary inﬁnitesimal functions, {ε

(x)} and {ε

µν

(x)}.Itisconvenienttoregard

µν

(x)andξ

(x) ≡ iε

(x)x

+ ε

(x),

as the 10 independent inﬁnitesimal functions. Such a choice avoids the explicit

appearance of x

. Furthermore, we can always choose ε

(x)suchthat

(x) = 0andε

µν

(x) = 0,

so that the coordinate and ﬁeld transformations are completely separated.

Based on this fact, we use the Latin indices for ε

(x) and the Greek indices for

and x

. The Latin indices, i, j, k, ...,alsoassumethevalues0,1,2,and3.Then

the transformations under consideration are

δx

= ξ

(x)andδφ(x) =

iε

(x)S

φ(x), (10.9.65)

φ(x) =−ξ

(x)∂

φ(x) +

iε

(x)S

φ(x). (10.9.66)

10.9 Weyl’s Gauge Principle 431

This notation emphasizes the similarity of the ε

(x) transformations to the linear

transformations of Weyl’s gauge principle. Actually, in Utiyama’s gauge principle,

the ε

(x) transformations alone are considered in the local 6-parameter Lorentz

transformation . The ξ

(x) transformations correspond to the general coordinate

transformation.

According to the convention we just employed, the differential operator ∂

must

have a Greek index. In the matter Lagrangian density L

matter

,wethenhavethetwo

kinds of indices, and we shall regard L

matter

as a given function of φ(x)and

∂

φ(x),

satisfying the identities, (10.9.59) and (10.9.60). The original matter Lagrangian

density L

matter

is obtained by setting

∂

φ(x) = δ

∂

φ(x).

The matter Lagrangian density L

matter

is not invariant under the local 10-parameter

transformations, (10.9.65) or (10.9.66), but we shall later obtain an invariant

expression by replacing

∂

φ(x) with a suitable covariant derivative D

φ(x)inthe

matter Lagrangian density L

matter

The transformation of ∂

φ(x)isgivenby

δ∂

φ =

iε

∂

φ +

i(∂

φ − (∂

)(∂

φ), (10.9.67)

and the original matter Lagrangian density L

matter

transforms according to

δL

matter

≡−(∂

−

i(∂

We note that it is J

instead of T

that appears here. The reason for this is that we

have not included the extra term (∂

δx

matter

in Eq,(10.9.56). The left-hand side

of Eq. (10.9.56) actually has the value

δL

matter

+ (∂

δx

) L

matter

≡−(∂

−

i(∂

We shall now look for the modiﬁed matter Lagrangian density L



matter

which

makes the matter action functional I

matter

[φ] invariant under (10.9.65) or (10.9.66).

The extra term just mentioned is of a different kind in that it involves L

matter

and

not ∂L

matter

∂(

∂

φ). In particular, the extra term includes the contributions from

terms in L

matter

which do not contain the derivatives. Thus it is clear that we cannot

remove the extra term by replacing the derivative

∂

with a suitable covariant

derivative D

. For this reason, we shall consider the problem in two stages. First,

we eliminate the noninvariance arising from the fact that ∂

φ(x) is not a covariant

quantity, and second, we obtain an expression L



matter

satisfying

δL



matter

≡ 0. (10.9.68)

432 10 Calculus of Variations: Applications

Because the invariance condition (10.9.56) for the matter action functional I

matter

requires the matter Lagrangian density L



matter

to be an invariant scalar density

rather than an invariant scalar, we shall make a further modiﬁcation, replacing



matter

with L



matter

, which satisﬁes

δL



matter

+ (∂



matter

≡ 0. (10.9.69)

The ﬁrst part of this program can be accomplished by replacing

∂

φ in L

matter

with a covariant derivative D

φ which transforms according to

δ(D

φ) =

iε

φ) −iε

φ). (10.9.70)

The condition (10.9.68) follows from the identities, (10.9.59) and (10.9.60). To do

this, it is necessary to introduce 40 new ﬁeld variables toward the end. We ﬁrst

consider the ε

transformations, and eliminate the ∂

term in (10.9.67) by setting

φ ≡ ∂

φ +

φ, (10.9.71)

where A

with

=−A

are24newﬁeldvariables.

Wecanthenimposethecondition

δ(D

φ) =

iε

φ) −(∂

)(D

φ), (10.9.72)

which determines the transformation properties of A

uniquely. They are

δA

=−∂

+ε

+ ε

− (∂

. (10.9.73)

The position of the last term in Eq. (10.9.67) is rather different. The term

involving ∂

is inhomogeneous in the sense that it contains φ rather than ∂

φ,

but this is not true for the last term. Correspondingly, the transformation law for

φ, (10.9.72), is already homogeneous. This means that to force the covariant

derivative D

φ to transform according to Eq. (10.9.70), we shall add to D

φ not a

term in φ but rather a term in D

φ itself. In other words, we merely multiply by a

new ﬁeld,

φ ≡ e

φ. (10.9.74)

Here, the e

are 16 new ﬁeld variables with the transformation properties deter-

mined by Eq. (10.9.70) to be

δe

= (∂

− iε

. (10.9.75)

10.9 Weyl’s Gauge Principle 433

We note that the ﬁelds e

and A

are independent and unrelated at this stage,

though they will be related by the Euler–Lagrange equations of motion.

We ﬁnd the invariant matter Lagrangian density L



matter

deﬁned by



matter

≡ L

matter

(φ, D

φ),

which is an invariant scalar. We can obtain the invariant matter Lagrangian density



matter

, which is an invariant scalar density by multiplying L



matter

by a suitable

function of the new ﬁeld variables,



matter

≡ EL



matter

= EL

matter

(φ, D

φ).

The invariance condition (10.9.69) for L



matter

is satisﬁed if a factor E itself is an

invariant scalar density,

δE + (∂

)E ≡ 0.

The only function of the new ﬁeld variable e

which obeys this transformation law

and does not involve the derivatives is

E = [det(e

)]

−1

, (10.9.76)

where the arbitrary constant factor has been chosen such that E = 1whene

set equal to δ

. The ﬁnal form of the modiﬁed matter Lagrangian density L



matter

which is an invariant scalar density is given by



matter

(φ, ∂

φ, e

, A

) ≡ EL

matter

(φ, D

φ). (10.9.77)

As in the case of Weyl’s gauge principle, we can deﬁne the modiﬁed current

densities in terms of L

matter

(φ, D

φ)by

≡

∂L



matter

∂e

≡ Ee



∂L

matter

∂(D

φ)

φ − δ

matter



, (10.9.78)

≡−2

∂L



matter

∂A

≡ iEe

∂L

matter

∂(D

φ)

φ, (10.9.79)

where e

is the inverse of e

, satisfying

= δ

, e

= δ

. (10.9.80)

In order to express the conservation laws of these currents in a simple form, we

extend the deﬁnition of D

φ. Originally, it was deﬁned for φ(x), and is to be

deﬁned for any other quantity which is invariant under the ξ

transformations

434 10 Calculus of Variations: Applications

and transforms linearly under the ε

transformations . We extend D

to any

quantity which transforms linearly under the ε

transformations by ignoring the

transformations altogether. Thus we have

≡ ∂

− A

iν

, (10.9.81)

according to the ε

transformation law of e

. We call this the ε covariant derivative.

We calculate the commutator of the ε covariant derivatives as,

, D

]φ =

µν

φ, (10.9.82)

where R

jµν

is deﬁned by the following equation:

jµν

≡ ∂

jµ

− ∂

jν

−A

kµ

jν

+ A

kν

jµ

. (10.9.83)

This quantity is covariant under the ε

transformations. R

jµν

is closely analogous to

the ﬁeld strength tensor F

αµν

ofthenon-Abeliangaugeﬁeld.R

µν

is antisymmetric

in both pairs of the indices.

In terms of the ε covariant derivative, the ten conservation laws of the currents,

(10.9.78) and (10.9.79), are expressed as

) + T

) = S

µν

, (10.9.84)

= T

iµ

− T

jµ

. (10.9.85)

Now we examine our ultimate goal, the Lagrangian density L

of the ‘‘free’’

self-interacting gravitational ﬁeld. We examine the commutator of D

and D

acting

on φ(x). After some algebra, we obtain

, D

]φ =

φ − C

φ, (10.9.86)

where

≡ e

µν

, C

≡ (e

− e

. (10.9.87)

We note that the right-hand side of Eq. (10.9.86) is not simply proportional to φ but

also involves D

φ.

The Lagrangian density L

for the ‘‘free’’ self-interacting gravitational ﬁeld must

be an invariant scalar density. If we set L

= EL

,thenL

must be an invariant

scalar and a function only of the covariant quantities R

and C

.Alltheindicesof

these expressions are of the same kind, unlike the case of the non-Abelian gauge

ﬁeld, so that we can take the contractions of the upper indices with the lower

indices.

10.9 Weyl’s Gauge Principle 435

The requirement that L

is an invariant scalar in two separate spaces is reduced

to the requirement that it is an invariant scalar in one space. We have a linear

invariant scalar which has no analog in the case of the non-Abelian gauge ﬁeld,

namely, R ≡ R

. There exists a few quadratic invariants, but we choose the lowest

order invariant. Thus we are led to the Lagrangian density L

for the ‘‘free’’

self-interacting gravitational ﬁeld,

2κ

ER, (10.9.88)

which is linear in the derivatives. In Eq. (10.9.88), κ is Newton’s gravitational

constant.

So far, we have given neither any geometrical interpretation of the local ten-

parameter Poincar

e transformation , (10.9.65), nor any interpretation of the 40 new

ﬁelds, e

(x)andA

(x). We shall now establish the connection of the present theory

with the standard metric theory of the gravitational ﬁeld.

Under the ξ

transformation which is a general coordinate transformation,

(x) transforms like a contravariant vector, while e

(x)andA

(x) transform like

covariant vectors. Then the quantity

µν

(x) ≡ e

(x)e

kν

(x) (10.9.89)

is a symmetric covariant tensor, and therefore may be interpreted as the metric

tensor of a Riemannian space. It remains invariant under the ε

transformations.

We shall abandon the convention that all the indices are to be lowered or raised by

the ﬂat-space metric η

µν

,andweuseg

µν

(x) instead as the metric tensor. We can

easily show that

E =

−g(x)withg(x) ≡ det(g

µν

(x)). (10.9.90)

From Eq. (10.9.89), we realize that e

(x)ande

(x) are the contravariant and

covariant components of a tetrad system in Riemannian space. The ε

transforma-

tions are the tetrad rotations. The Greek indices are the world tensor indices and

the Latin indices are the local tensor indices of this system. The original generic

matter ﬁeld φ(x) may be decomposed into local tensors and local spinors. From

the local tensors, we can form the corresponding world tensors by multiplying by

(x)ore

(x).

For example, from a local vector v

(x), we can form the world vector as

(x) = e

(x)v

(x)andv

(x) = e

(x)v

(x). (10.9.91)

We note that

(x) = g

µν

(x)v

(x),

436 10 Calculus of Variations: Applications

so that Eq. (10.9.91) is consistent with the deﬁnition of the metric g

µν

(x),

Eq. (10.9.89).

The ﬁeld A

jµ

(x) is regarded as a local afﬁne connection with respect to the tetrad

system since it speciﬁes the covariant derivatives of local tensors or local spinors.

For a local vector, we have

(

= ∂

jν

= ∂

−A

jν

(10.9.92)

We notice that the relationship between D

φ and D

φ, (10.9.74), could be

written simply as

φ = D

φ, (10.9.93)

according to the convention (10.9.91). We shall, however, make a distinction

between D

and D

for a later purpose. We deﬁne the covariant derivative of

a world tensor in terms of the covariant derivative of the associated local tensor.

Thus, we have

(

≡ e

= ∂

+ 

µν

≡ e

= ∂

− 

µν

(10.9.94)



µν

≡ e

≡−e

. (10.9.95)

We note that this deﬁnition of 

µν

is equivalent to the requirement that the

covariant derivative of the tetrad components shall vanish,

(

≡ 0,

≡ 0.

(10.9.96)

For a generic quantity α transforming according to

δα =

iε

α + (∂

)

α, (10.9.97)

the covariant derivative of α is deﬁned by

α ≡ ∂

α +

α + 

µν



α. (10.9.98)

The ε covariant derivative of α, deﬁned by Eq. (10.9.81), is obtained by simply

dropping the last term in (10.9.98). We calculate the commutator of the covariant

derivative of α with the result,

, D

]α =

µν

α + R

σµν



α − C

µν

α,

10.10 Path Integral Quantization of Gauge Field I 437

where R

σµν

and C

µν

are deﬁned in terms of R

jµν

and C

in the usual way. These

quantities are the world tensors and can be expressed in terms of 

µν

in the form

σµν

= ∂



σµ

− ∂



σν

− 

λµ



σν

+ 

λν



σµ

, C

µν

= 

µν

− 

νµ

(10.9.99)

We see that R

σµν

is the Riemann tensor formed from the afﬁne connection 

µν

From Eq. (10.9.96), we have D

µν

(x) ≡ 0. It is consistent to interpret 

µν

as an

afﬁne connection in a Riemannian space. The deﬁnition of 

µν

, Eq. (10.9.95),

does not guarantee that it is symmetric so that it is not the Christoffel symbol in

general. In the absence of the matter ﬁeld, however, 

µν

is symmetric so that it

is the Christoffel symbol. The curvature scalar has the usual form, R ≡ R

,where

µν

≡ R

µλν

. The Lagrangian density L

for the ‘‘free’’ self-interacting gravitational

ﬁeld , Eq. (10.9.88), is the usual one,

µν

(x), 

µν

(x)) =

2κ

−gg

µν

(∂



µλ

− ∂



µν

+

µλ



νρ

− 

µν



λρ

(10.9.100)

10.10

Path Integral Quantization of Gauge Field I

In this section, we discuss the path integral quantization of the gauge ﬁeld.Whenwe

apply the path integral representation of



0, out



0, in





0, out



0, in





D[φ

]exp





x L(φ

(x), ∂

(x))



naively to the path integral quantization of the gauge ﬁeld, we obtain the results

that violate the unitarity. The origin of this difﬁculty lies in the gauge invariance of

the gauge ﬁeld action functional I

gauge

αµ

], i.e., the four-dimensional transversality

of the kernel of the quadratic part of the gauge ﬁeld action functional I

gauge

αµ

In another word, when we path-integrate along the gauge equivalent class, the

functional integrand remains constant and we merely calculate the orbit volume

. We introduce the hypersurface (the gauge-ﬁxing condition),

βµ

(x)) = 0, α = 1, ..., N,

in the manifold of the gauge ﬁeld which intersects with each gauge equivalent

class only once and perform the path integration and the group integration on this

hypersurface. We shall complete the path integral quantization of the gauge ﬁeld

and the separation of the orbit volume V

simultaneously. This method is called the

Faddeev–Popov method. Next we generalize the gauge-ﬁxing condition to the form

βµ

(x)) = a

(x), α = 1, ..., N,