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Path Integrals and Quantum Anomalies$

Kazuo Fujikawa and Hiroshi Suzuki

Print publication date: 2004

Print ISBN-13: 9780198529132

Published to British Academy Scholarship Online: September 2007

DOI: 10.1093/acprof:oso/9780198529132.001.0001

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(p.253) APPENDIX B

(p.253) APPENDIX B

Source:
Path Integrals and Quantum Anomalies
Publisher:
Oxford University Press

FIELD THEORY IN CURVED SPACE-TIME

In this appendix we briefly summarize the basic properties of field theory in curved space-time.

B.l Coordinate transformation and energy-momentum tensor

The Lagrangian of Einstein's general theory of relativity in the presence of QCD-type gauge theory is defined by

(B.1)
L = R 2 κ + ψ ¯ i e κ μ γ κ ( μ i 2 A μ m n σ m n i A μ a T a ) ψ m ψ ¯ ψ 1 4 g μ ν g α β F μ α a F ν β a
where k = 8πG with the Newton constant G. The Greek indices which describe the Minkowski coordinates of the curved space and the Roman indices which describe the Lorentz coordinates of the flat space attached to each point of the curved space are related to each other by the vierbein e k μ ( x ) . The metric of the flat Lorentz frame is denoted as G mn. The vierbein is connected to the metric by
(B.2)
e m μ e n μ = G m n , G m n = ( 1 , 1 , 1 , 1 ) , g μ ν = e m μ e m ν , g μ ν g ν ρ = δ μ ρ , d s 2 = g μ ν d x μ d x ν .
The raising and lowering of the Minkowski indices μ is realized by the metric gμν(x) and the raising and lowering of the flat Lorentz indices m is realized by G mn. The basic symmetry of Einstein's theory, namely, the general coordinate transformation, does not admit a double-valued representation such as the rotation group for spin 1/2 in flat space, and thus we represent the Dirac field as a double-valued representation of the Lorentz group of the flat space attached to each point of the curved space-time.

The covariant derivative in the curved space is generally given by

(B.3)
D μ = μ i 2 A μ m n S m n i Γ β μ α U α β i A μ a T a
and contains the generators S mn of the Lorentz group and the generators U α β of the general coordinate transformation GL(4, R) in addition to the generators T a of the ordinary gauge group. These generators are defined by
(B.4)
[ S m n , S κ l ] = i ( G m κ S n l + G n l S m κ G m l S n κ G n κ S m l ) , [ U α β , U γ δ ] = i ( δ α δ U γ β δ γ β U α δ ) .
(p.254) To be more explicit, the generators for spin 1/2 and 1 are, respectively, given by
(B.5)
S m n ( 1 / 2 ) = σ m n = i 4 [ γ m , γ n ] , ( S m n ( 1 ) ) k l = i ( δ m k δ n l δ m l δ n k ) .
On the other hand, the generators U α β of the general coordinate transformation are given for the covariant vector A ν and the contravariant vector A ν respectively by
(B.6)
( U α β ) μ ν A ν = ( i δ α ν δ α β ) A ν , ( U α β ) μ ν A ν = ( i δ ν μ δ ν β ) A ν .
Consequently, the covariant derivatives are defined by
(B.7)
D μ A ν = μ A ν + Γ ρ μ ν A ρ , D μ A ν = μ A ν Γ ν μ ρ A ρ .

The gauge field A μ m n appearing in the covariant derivative is called the spin connection and the gauge field Γ β μ α is called the affine connection, respectively, and these are expressed in terms of e κ μ and g μν as

(B.8)
A μ m n = 1 2 e m λ e n ρ ( C λ ρ μ C ρ λ μ C μ λ ρ ) , C λ ρ μ e λ κ ( ρ e κ μ μ e κ ρ ) , Γ μ ν λ { λ μ ν } = 1 2 g λ ρ ( μ g ν ρ + ν g μ ρ ρ g μ ν ) .
The fields A μ m n and Γ β μ α in the present formulation, in which the torsion freedom is ignored, are also called respectively the Ricci rotation coefficient and the Christoffel symbol.

By recalling the basic relation of Riemann which states that the geometrical length is independent of the choice of coordinate systems

(B.9)
d s 2 = g μ ν ( x ) d x μ d x ν = g μ ν ( x ) d x μ d x ν ,
the general coordinate transformation laws of the quantities with Minkowski indices are given by
(B.10)
g μ ν ( x ) = x α x μ x β x ν g α β ( x ) , e κ μ ( x ) = x μ x ν e κ ν ( x ) , ψ ( x ) = ψ ( x ) , A μ ( x ) = x α x μ A α ( x ) .
For an infinitesimal transformation
(B.11)
x μ = x μ ξ μ ( x ) , Λ β α ( x ) β ξ α ( x )
the general coordinate transformation is expressed in terms of the generators as (p.255)
(B.12)
e κ μ ( x ) = [ exp ( i U α β Λ β α ) e κ ] μ ( x ) = e κ μ ( x ) Λ ν μ ( x ) e κ ν ( x ) .
The transformation law of the metric tensor is derived from the transformation law of the vierbein as
(B.13)
g μ ν ( x ) = e κ μ ( x ) e k ν ( x ) = g μ ν ( x ) Λ λ μ g λ ν ( x ) Λ λ ν g μ λ ( x )
and the transformation law of the affine connection is derived from its definition as
(B.14)
Γ β μ α ( x ) = Γ β μ α ( x ) + Λ β λ Γ λ μ α ( x ) Λ λ α Γ β μ λ ( x ) + Λ μ λ Γ β λ α ( x ) + μ Λ β α ( x ) .

The infinitesimal local Lorentz transformation is given in terms of the parameter ωmn(x) as

(B.15)
ψ ( x ) = exp [ i 2 S m n ( 1 / 2 ) ω m n ( x ) ] ψ ( x ) = ψ ( x ) + i 2 σ m n ω m n ( x ) ψ ( x ) e k μ ( x ) = exp [ i 2 S m n ( 1 ) ω m n ( x ) ] e k μ ( x ) = e k μ ( x ) ω k n ( x ) e n μ ( x ) A μ m n ( x ) = A μ m n ( x ) + A μ m l ( x ) ω l n ( x ) + A μ n l ( x ) ω m l ( x ) + μ ω m n ( x ) .
by using the generators of the Lorentz group, just as the ordinary gauge transformations. The transformation law of the spin connection in the last expression is obtained from the transformation of the covariant derivative
(B.16)
D μ = μ i 2 A μ m n S m n = exp ( i 2 S m n ω m n ) D μ exp ( i 2 S m n ω m n )
or from the definition of A μ m n in terms of e κ μ and the transformation law of e κ μ .

Einstein's theory contains two gauge fields e κ μ and A μ m n , but both of these fields are expressed in terms of the vierbein e κ μ . As the gauge freedom, the vierbein contains the localized Poincaré transformations with 10 parameters, namely, four localized translations ξμ(x) and six localized Lorentz transformations ωmn(x). In the present formulation, the condition called the metric condition

(B.17)
( D μ e ) k ν = μ e k ν + Γ β μ ν e k β + A μ k l e l ν = 0
is automatically satisfied. This metric condition implies that the inner product of two vectors parallel transported from a point x to the point x + dx is defined by the metric g μν(x + dx).

As an application of this metric condition, we can show that the vierbein e κ μ at an arbitrary point, for example, at the oriein, is chosen to be

(B.18)
e κ ν ( 0 ) = δ k ν , μ e k ν ( 0 ) = 0 .
This choice of the vierbein is used in the body of the present book to evaluate quantum anomalies. The proof of this choice of the vierbein proceeds as follows: (p.256) We first apply a coordinate transformation x ′μ = a ν μ x ν to the vierbein e κ μ by-choosing a suitable constant a ν μ near the origin such that
(B.19)
e k μ ( 0 ) = a ν μ e k ν ( 0 ) = δ k μ .
We next apply a specific infinitesimal coordinate transformation near the origin
(B.20)
ξ α ( x ) = 1 2 Γ β μ α ( 0 ) x β x μ
such that Γ β μ α ( 0 ) = 0 without changing e k μ ( 0 ) . Finally, we apply a local Lorentz transformation parametrized by ω m n ( x ) = A μ m n ( 0 ) x μ so that A μ m n ( 0 ) = 0 without changing e k μ ( 0 ) . When one uses the two relations Γ β μ α ( 0 ) = 0 and A μ m n ( 0 ) = 0 in the metric condition (B.17), we obtain the desired result.

The metric condition shows that we can freely commute the covariant derivative with e k μ and g μν. For example, when one applies the commutator of covariant derivatives to an arbitrary covariant vector A ρ = e ρ n A n , one obtains

(B.21)
R α ρ μ ν A α ( [ D μ , D ν ] A ) ρ
(B.22)
= e ρ n ( [ D μ , D ν ] A ) n = e m α e n ρ R m n μ ν A α .
This relation shows that the Riemann–Christoffel curvature tensor evaluated in terms of the affine connection Γ β μ α .
(B.23)
R α ρ μ ν μ Γ ρ μ α ν Γ ρ μ α + Γ λ μ α Γ ρ ν λ Γ λ ν α Γ ρ μ λ
agrees with the curvature tensor R m n μ ν evaluated in terms of the spin connection A μ m n .

As an important local symmetry of the matter and gauge fields in gauge theory, we have the Weyl symmetry defined by

(B.24)
e k μ ( x ) exp [ α ( x ) ] e k μ ( x ) , e k μ ( x ) exp [ α ( x ) ] e k μ ( x ) , g μ ν ( x ) exp [ 2 α ( x ) ] g μ ν ( x ) , g μ ν ( x ) exp [ 2 α ( x ) ] g μ ν ( x ) ψ ( x ) exp [ 3 2 α ( x ) ] ψ ( x ) , A μ a ( x ) A μ a ( x ) , i 2 A μ m n ( x ) σ m n i 2 A μ m n ( x ) σ m n 1 2 [ μ α ( x ) γ μ γ λ λ α ( x ) ]
where the transformation law of the spin connection A μ m n is derived from the transformation law of the vierbein e k μ , since A μ m n is expressed in terms of e k μ . The Weyl transformation changes the length as
(B.25)
d s 2 = g μ ν d x μ d x ν exp [ 2 α ( x ) ] d s 2 = exp [ 2 α ( x ) ] g μ ν d x μ d x ν
but the local angle is preserved, and for this reason it is also called a conformal transformation. When one defines g = det g μν, it is confirmed that the matter (p.257) part of the action d 4 x g L in terms of the Lagrangian (B.l) is invariant under the Weyl transformation if one sets the fermion mass m = 0. The action for a massless scalar theory is also rendered invariant under the Weyl transformation if one chooses L = 1 2 g μ ν μ φ ν φ + 1 6 R φ 2 . However, the action of Einstein's gravitational theory itself is not Weyl invariant.

The scalar curvature R in the Einstein action, which is also called the Einstein–Hilbert action, and also the Ricci tensor R μν are defined in terms of the curvature tensor by

(B.26)
R = g μ ν R μ ν , R ρ ν R α ρ α ν .

When one considers the variation with respect to the vierbein e k μ ( x ) in the Einstein–Hilbert action (B.l), the Einstein equation

(B.27)
R μ ν 1 2 g μ ν R = κ T μ ν ( x )
is obtained. The (naive) energy-momentum tensor T μν(x) generated by the matter fields in this equation is defined by59
(B.28)
T μ ν ( x ) 1 g e κ μ ( x ) δ δ e k ν ( x ) S matter = 1 2 [ ψ ¯ i γ μ D ν ψ ( D ν ψ ¯ ) i γ μ ψ ] g α β F μ α F ν β g μ ν L matter + 1 4 e k μ e l ν e m ρ ε k l m n D ρ ( ψ ¯ γ n γ 5 ψ )
where we symmetrized the expression with respect to the replacement of ψ and ψ ¯ . We also normalized the completely anti-symmetric symbol ε1230 = 1 and used the property of the Dirac matrix {γk,S mn} = εkmnlγlγ5.

To stuay tne symmetry requirement on the energy-momentum tensor, we recall the basic property of the Einstein-Hilbert action. We first note the following relation

(B.29)
δ d 4 x g R = d 4 x g δ g μ ν ( R μ ν 1 2 g μ ν R ) .
valid for an arbitrary variation δg μν(x) of the metric tensor. As a specific variation of the metric, we consider the variation induced by a coordinate transformation
(B.30)
δ g μ ν ( x ) g μ ν ( x ) g μ ν ( x ) = ξ λ λ g μ ν ( x ) λ ξ μ g λ ν ( x ) λ ξ ν g λ μ ( x ) = g μ λ D λ ξ ν g ν λ D λ ξ μ .
(p.258) When one recalls that the Einstein–Hilbert action is invariant under the coordinate transformation and that the covariant derivative commutes with the metric, the above relation (B.29) valid for an arbitrary variation implies
(B.31)
d 4 x g ξ ν D μ ( R μ ν 1 2 g μ ν R ) = 0 .
Since ξν(x) is arbitrary we conclude
(B.32)
D μ ( R μ ν 1 2 g μ ν R ) = 0 .
When one combines this relation with the Einstein equation (B.27), one concludes the requirements on the classical energy-momentum tensor
(B.33)
D μ T μ ν = 0 , T μ ν = T ν μ .

B.2 Path integral measure in gravitational theory

It is necessary to define the path integral measure carefully when one analyzes the quantum theory of gravity and the quantization of fields in curved space-time in the path integral formulation. In particular, the treatment of general coordinate transformations is subtle and a naive treatment could induce spurious anomalies. Also the Weyl anomaly is uniquely specified only when one uses the general coordinate invariant measure. In the following, we discuss the definition of the path integral measure which does not induce spurious breaking of general coordinate transformations.

It is convenient to use the BRST symmetry in this analysis, though the BRST symmetry itself is somewhat technical. The basic idea of the BRST transformation is based on writing the parameter ξμ(x) of general coordinate transformations in the form ξμ(x) = iλc μ(x) where λ is a constant real Grassmann parameter and c μ(x) is the Grassmann variable called the Faddeev-Popov ghost field. When one uses the BRST superfield notation, one writes the original field variable together with its variation under the BRST transformation by using a constant Grassmann parameter θ as

(B.34)
e k μ ( x , θ ) = e k μ ( x ) + i θ [ c ρ ρ e k μ ( x ) ρ c μ e k ρ ( x ) ] g μ ν ( x , θ ) = g μ ν ( x ) + i θ [ c ρ ρ g μ ν ( x ) + μ c ρ g ρ ν ( x ) + ν c ρ g μ ρ ( x ) ] g ( x , θ ) det [ g μ ν ( x , θ ) ] = g ( x ) + i θ [ c ρ ρ g ( x ) + 2 ρ c ρ g ( x ) ] ψ ( x , θ ) = ψ ( x ) + i θ c ρ ρ ψ ( x ) c μ ( x , θ ) = c μ ( x ) + i θ c ρ ρ c μ ( x ) c ¯ μ ( x , θ ) = c ¯ μ ( x ) + θ B μ ( x )
where we used the transformation law of ψ, δψ(x) = ψ′(x) − ψ(x) = ξρρψ(x), and wrote the anti-ghost field c ¯ μ ( x ) together with its variation, the auxiliary field B μ(x).

(p.259) The BRST transformations are defined as a translation in the Grassmann parameter θ → θ + λ. Consequently, the first component of the superfield, which does not contain the parameter θ, varies by an amount proportional to the parameter λ and the second component, which contains the parameter θ, remains invariant. To be explicit, the transformation law δλ of the ghost field c μ(x) is given by (noting that both c μ(x) and λ are Grassmann numbers)

(B.35)
δ λ c μ ( x ) = i λ c ρ ρ c μ ( x ) , δ λ [ c ρ ρ c μ ( x ) ] = i λ [ c α α c ρ ( x ) ] ρ c μ ( x ) + c ρ ρ [ i λ c α α c μ ( x ) ] = i λ [ c α α c ρ ρ c μ ( x ) c ρ ρ c α α c μ ( x ) ] = 0.
Namely, the first component receives a transformation proportional to the second component, and if one uses its result the second component becomes invariant. The BRST transformation is thus consistently defined. The treatment of the anti-ghost is exceptional, and its variation is defined by δ λ c ¯ μ ( x ) = λ B ( x ) , δ λ B ( x ) = 0 and, as a result the measure D c ¯ μ D B μ is BRST invariant. When one uses the notation of the BRST superfield, the gauge condition δμ g μν = 0, for example, is defined by collecting the terms linear in θ in the expression
(B.36)
c ¯ ν ( x , θ ) μ g μ ν ( x , θ )
which defines both the gauge fixing and compensating terms simultaneously.

In passing we note that the transformation law of the ghost (B.34) does not appear as a transformation law of a contravariant vector quantity, but if one considers the differential of the superfield itself one obtains

(B.37)
d c μ ( x , θ ) = d c μ ( x ) + i θ [ c ρ ρ d c μ ( x ) ρ c μ d c ρ ( x ) ]
which corresponds to the replacement. ξρ(x) → iθc ρ(x) in the general coordinate transformation of the contravariant vector dc μ(x). This property plays a fundamental role when we discuss the BRST invariant path integral measure.

We now discuss the path integral measure which is invariant under the BRST symmetry associated with general coordinate transformations. We start with the simplest fields which do not carry the Minkowski indices such as the fermions ψ(x) and ψ ¯ ( x ) . We define the weight 1/2 fields60

(B.38)
ψ ( x ) ( g ) 1 / 4 ψ ( x ) , ψ ¯ ˜ ( x ) ( g ) 1 / 4 ψ ¯ ( x )
and examine the path integral of the mass term of the fermion
(B.39)
D ψ ¯ ˜ D ψ ˜ exp [ i m d 4 x g ψ ¯ ( x ) ψ ( x ) ] = D ψ ¯ ˜ D ψ ˜ exp [ i m d 4 x ψ ¯ ˜ ( x ) ψ ˜ ( x ) ] .
In this relation, the action on the left-hand side is invariant under the general coordinate (or associated BRST) transformation and the right-hand side is a (p.260) constant independent of the metric variable. We thus conclude that the path integral measure on the left-hand side is invariant under the general coordinate (or BRST) transformation. One can in fact confirm that the Jacobian for the BRST transformation becomes trivial, i.e., 1, independently of the choice of basis sets, and thus free of general coordinate anomalies. See eqn (7.21).

We now generalize this analysis to the cases such as the vector field A α(x) and the second-rank tensor field A αβ(x). We first convert these fields into fields without the Minkowski indices A a ( x ) = e α a A α ( x ) and A a b ( x ) = e a α e b β A α β ( x ) by using the vierbein. We then consider the weight 1/2 field prescription as above

(B.40)
D A ˜ a D [ ( g ) 1 / 4 e α a A α ] = Π x { [ g ( x ) ] n / 4 det e α a ( x ) } D A α = D [ ( g ) k A α ] , k = n + 2 4 n , D A ˜ a D [ ( g ) 1 / 4 e a α A α ] = D [ ( g ) k A α ] , k = n 2 4 n , D A ˜ a b D [ ( g ) 1 / 4 e α a e β b A α β ] = D [ ( g ) k A α β ] , k = n + 4 4 n , D A ˜ a b D [ ( g ) 1 / 4 e a α e b β A α β ] = D [ ( g ) k A α β ] , k = n 4 4 n ,
which define the general coordinate (or associated BRST) invariant path integral measure. When we give two definitions of the measure in eqn (B.40), the latter definition is obtained from the first one by first extracting the vierbein as the Jacobian and then distributing the Jacobian to all the degrees of freedom equally. This second definition may not be said to be precise, but we cannot convert the metric g μν into a quantity without the Minkowski indices by multiplying the vierbein. The second definition of the measure works for the case of the metric tensor also. We also give the definitions for the vector and tensor fields in arbitrary d = n dimensional space-time in eqn (B.40) for the convenience of applications in the present book.

In this way the path integral measure invariant under the BRST symmetry associated with the general coordinate transformation is given by

(B.41)
d μ = d μ ( g α β ) D [ ( g ) 1 / 4 e μ a c μ ] D c ¯ μ D B μ D ψ ¯ ˜ D ψ ¯ .
The part containing the metric is defined by d μ ( g α β ) = D [ ( g ) k g α β ] as we explained above. The measure for the ghost variable is determined by c ¯ μ = ( g ) 1 / 4 e μ a c μ by using the fact that the differential of the ghost variable is transformed as a contravariant vector under the BRST transformation. In this path integral measure the BRST invariance of the part d μ ( g α β ) D c ˜ μ is not manifest, but its invariance is confirmed by showing that the result after BRST transformation agrees with the result before the BRST transformation as (p.261)
(B.42)
d μ [ g α β ( x , θ ) ] D c ˜ μ ( x , θ ) = d μ [ g α β ( x , θ ) ] det { [ g ( x , θ ) ] 1 / 4 e μ a ( x , θ ) } 1 D c μ ( x , θ ) = d μ [ g α β ( x ) ] det { [ g ( x , θ ) ] 1 / 4 e μ a ( x , θ ) } 1 D c μ ( x , θ ) = d μ [ g α β ( x ) ] D { [ g ( x ) ] 1 / 4 e μ a ( x ) c μ ( x ) } .
The above proof proceeds as follows: The measure for the metric is invariant for a fixed c μ and thus one can set θ = 0 in g αβ(x,θ) in the third line. We next fix the metric, and then the Jacobian, det { [ g ( x , θ ) 1 / 4 e μ a ( x , θ ) ] } 1 , is the one which renders a general (fermionic) contravariant vector invariant and thus when combined with D c μ ( x , = ) we can set θ = 0. It is important to keep in mind that one needs to consider the metric and the ghost variables always together in the analysis of the gravitational path integral measure.

Notes:

(59) In deriving this expression, it is convenient to note the relation δ A μ m n = ( 1 / 2 ) e m λ e n ρ ( δ C λρμ δ C ρλμ δ C μλρ ) where δ C λρμ = e λ k ( D ρ δ e k μ D μ δ e k ρ ) . This relation follows from the fact that the difference of two spin connections behaves as a tensor.

(60) The weight 1/2 fields are defined as the field variables (without the Minkowski indices) multiplied by the weight factor (−g)1/4, and these variables transform under general coordinate transformations as in eqn (7.19).