Comments on the Stress-Energy Tensor Operator in Curved Spacetime

On quantum ground, 〈 ˆ φ ( x ) P ˆ φ ( x ) 〉 ω = 0, provided the left-hand side is defined via the point-splitting procedure.

“On quantum ground things dramatically change since 〈 ˆ ϕ ( x ) P ˆ ϕ ( x ) 〉 ω = 0, provided the left-hand side is defined via point-splitting procedure (see also [3] where the same remark appears in terms of local Wick polynomials).”

• cites [3] [3] S. Hollands and R. M. Wald, 'Local Wick polynomials and time ordered products of quantum fields in curved spacetime'. Commun. Math. Phys. 223 , 289 (2001) [gr-qc/0103074] - sourceDisputed

  Section 4.1 (page unavailable) says that making every local Wick product containing the field-equation factor vanish would be a reasonable additional requirement, but explicitly states that the paper’s construction does not satisfy it. Thus the paper does not make the claimed zero-valued point-splitting remark; it identifies a failure of that property for its construction. (LLM, conf=0.88)

  Checked by a reasoning model — may be wrong.

  Experimental — accuracy not yet measured

The technique based on a *-algebra of Wick products of field operators in curved spacetime, in the local covariant version proposed by Hollands and Wald, is generalized to define the stress-energy tensor operator in curved globally hyperbolic spacetimes.

“The technique based on a ∗ -algebra of Wick products of field operators in curved spacetime, in the local covariant version proposed by Hollands and Wald, is strightforwardly generalized in order to define the stress-energy tensor operator in curved globally hyperbolic spacetimes.”

• cites [3] [3] S. Hollands and R. M. Wald, 'Local Wick polynomials and time ordered products of quantum fields in curved spacetime'. Commun. Math. Phys. 223 , 289 (2001) [gr-qc/0103074] - sourceSupported

As remarked in [3], a straightforward definition of the stress-energy tensor operator from local Wick products of differentiated fields would produce a non-conserved stress-energy tensor.

“However, as remarked in [3] such a definition would produce a non conserved stress-energy tensor.”

• cites [3] [3] S. Hollands and R. M. Wald, 'Local Wick polynomials and time ordered products of quantum fields in curved spacetime'. Commun. Math. Phys. 223 , 289 (2001) [gr-qc/0103074] - sourceSupported

Dω has the property that every map h ↦ Wω,n(h), h ∈ D_n(M), defines a symmetric operator-valued distribution.

“D ω enjoys two relevant properties. (a) Every map h ↦→ ˆ W ω,n ( h ), h ∈ D n ( M ), defines a symmetric operator-valued distribution .”

• cites [3] [3] S. Hollands and R. M. Wald, 'Local Wick polynomials and time ordered products of quantum fields in curved spacetime'. Commun. Math. Phys. 223 , 289 (2001) [gr-qc/0103074] - sourceSupported

Dω has the property that the operator-valued distributions Wω,n may give rise to operators interpretable as products of field operators evaluated at the same event.

“(b) Those operator-valued distributions may give rise to operators which can be interpreted as products of field operators evaluated at the same event .”

• cites [3] [3] S. Hollands and R. M. Wald, 'Local Wick polynomials and time ordered products of quantum fields in curved spacetime'. Commun. Math. Phys. 223 , 289 (2001) [gr-qc/0103074] - sourceSupported

If ω and ω' are two quasifree Hadamard states, Wω(M,g) and Wω'(M,g) are isomorphic under a canonical *-isomorphism αω'ω, as shown in Lemma 2.1 in [3].

“However, if ω, ω ′ are two quasifree Hadamard states, W ω ( M, g ) , W ω ′ ( M, g ) are isomorphic (not unitary in general) under a canonical ∗ -isomorphism α ω ′ ω : W ω ( M, g ) → W ω ′ ( M, g ), as shown in Lemma 2.1 in [3].”

• cites [3] [3] S. Hollands and R. M. Wald, 'Local Wick polynomials and time ordered products of quantum fields in curved spacetime'. Commun. Math. Phys. 223 , 289 (2001) [gr-qc/0103074] - sourceSupported

The left-hand side of (25) coincides with the expression given in Theorem 4.1 of [20] when λ coincides with μ^{-1} up to a suitable positive constant factor.

“as shown in Theorem 4.1 of [20] provided (using ℏ = c = 1) λ coincides with µ -1 with a suitable positive constant factor.”

• cites [20] [20] V. Moretti, 'One-loop stress-tensor renormalization in curved background: The relation between zeta-function and point-splitting approaches, and an improved point-splitting procedure'. J. Math. Phys. 40 , 3843 (1999) [gr-qc/9809006]. - sourcePartly supported

  The Comments (p. 30) explicitly relate the length scale u to the mass scale µ by µ = 2e^−γ/u, supporting the claimed inverse relationship up to a positive constant. The paper establishes the point-splitting relation in Theorem 3.1 (p. 28), not Theorem 4.1, and our copy does not establish that the unspecified left-hand side of the citing paper’s equation (25) is the same expression. (LLM, conf=0.82)

  Experimental — accuracy not yet measured

The distribution Z(+)n does not depend on the choice of T.

“The distributio Z (+) n does not depend on the choice of T (see [12] and Appendix A3 of [16]).”

• cites [12] [12] B. S. Kay and R. M. Wald, 'Theorem on the uniqueness and thermal properties of stationary, nonsingular, quasifree states on space-times with a bifurcate Killing horizon'. Phys. Rept. 207 , 49 (1991). - sourceNot checked — no full text of the source
• cites [16] [16] H. Sahlmann and R. Verch, 'Microlocal spectrum condition and Hadamard form for vector-valued quantum fields in curved spacetime'. Rev. Math. Phys. 13 , 1203 (2001). [math-ph/0008029] - sourceCould not verify

  Appendix A.3 states that the distributions G(1) and G(2) do not depend on the time function t. The passages found for Z(+)n and the choice of T do not identify Z(+)n with those distributions or address its dependence on T, so the claim cannot be verified from our copy. (LLM, conf=0.70)

  Experimental — accuracy not yet measured

The coefficients Uk are symmetric, i.e., if x,y in C, Un(x,y)=Un(y,x).

“It is possible to show that the coefficients are symmetric, i.e., if x, y ∈ C , U n ( x, y ) = U n ( y, x ) [19] and thus, since s ( x, y ) = s ( y, x ), it also holds”

• cites [19] [19] V. Moretti, 'Proof of the symmetry of the off-diagonal Hadamard/Seeley-deWitt's coefficients in C ∞ Lorentzian manifolds by a 'local Wick rotation' '. Commun. Math. Phys. 212 , 165 (2000) [gr-qc/9908068]. - sourceSupported

The locality and covariance requirement is generalized to Wick products of differentiated quantum fields.

“In particular, the locality and covariance requirement is generalized to Wick products of differentiated quantum fields.”

The proposed stress-energy tensor operator is conserved.

“The proposed stress-energy tensor operator turns out to be conserved and reduces to the classical form if field operators are replaced by classical fields satisfying the equation of motion.”

The proposed stress-energy tensor operator reduces to the classical form if field operators are replaced by classical fields satisfying the equation of motion.

“The proposed stress-energy tensor operator turns out to be conserved and reduces to the classical form if field operators are replaced by classical fields satisfying the equation of motion.”

Theorem 3.1: The map Φ which associates with every globally hyperbolic spacetime (M,g) and every class of smooth contravariant vector fields {a(ij)} the distribution f ↦ :K1 φ ... Kn φ(f):, with each K_i defined using the fields a(ij), is a locally and covariant differentiated quantum field in one variable.

“Let Φ be the map which associates with every globally hyperbolic spacetime ( M, g ) and every class smooth contravariant vector field on M , { a ( ij ) } i =1 ,... ,n, j =0 ,...L i , the (abstract) distribution f ↦→ : K 1 ϕ · · · K n ϕ ( f ): , where f ∈ D ( M ) and each K i being defined in (32) using the fields a ( ij ) .”

• Also stated in: § 3 The stress-energy operator in terms of local Wick products. · p. 19 · plain

  “Φ is a locally and covariant differentiated quantum field in one variable.”

• Also stated in: § 3 The stress-energy operator in terms of local Wick products. · p. 19 · plain

  “We conclude this part by showing that the introduced differentiated local Wick polynomial are local and covariant.”

• AssumptionThe authors state this as an assumption: dim(M)=4 and V'≡0 in the Klein-Gordon operator.

The tensor field t is conserved for V′ ≡ 0.

“where the smooth symmetric tensor field t is independent from either the quantum state, λ and λ ′ , is conserved for V ′ ≡ 0 and it is built up, via standard tensor calculus, by employing the metric and the curvature tensors at z , m , ξ , V ′ ( z ) and their covariant derivatives at z .”

The conservation requirement ∇·T ≡ 0 is equivalent to the requirement that the right-hand side of (46) vanishes for all f ∈ D(M) and all smooth contravariant vector fields X on M.

“Therefore the conservation requirement ∇· T ≡ 0 is equivalent to the requirement that the right-hand side of (46) vanishes for all f ∈ D ( M ) and smooth contravariant vector fields X on M .”

• AssumptionThe authors state this as an assumption: dim(M)=4 and V'≡0 in the Klein-Gordon operator.

For ξ = ξD, if V ≡ 0 and m = 0, the trace of Tαβ(x) vanishes on field solutions.

“For ξ = ξ D , if V ≡ 0 and m = 0, the action of the field ϕ turns out to be invariant under local conformal transformations ( g ( x ) → λ ( x ) g ( x ), ϕ ( x ) → λ ( x ) 1 / 2 -D/ 4 ϕ ( x )) and the trace of T αβ ( x ) vanishes on field solutions by (5).”

The properties (b) and (c) respectively imply the relations in M×M which hold when x=y are not light-like related.

“Referring to 2.2 , the properties (b) and (c) respectively imply the relations in M × M”

• Also stated in: § Appendix B. · p. 26 · plain

  “which hold when x = y are not light-like related.”

Changing the scale λ → λ′ changes the averaged stress-energy tensor by a smooth symmetric tensor field t.

“where the smooth symmetric tensor field t is independent from either the quantum state, λ and λ ′ , is conserved for V ′ ≡ 0 and it is built up, via standard tensor calculus, by employing the metric and the curvature tensors at z , m , ξ , V ′ ( z ) and their covariant derivatives at z .”

〈 ˆ φ ( z ) P ˆ φ ( z ) 〉 ω = 〈 P ( ˆ φ ( z )) ˆ φ ( z ) 〉 ω.

“In particular 〈 ˆ ϕ ( z ) P ˆ ϕ ( z ) 〉 ω = 〈 P ( ˆ ϕ ( z )) ˆ ϕ ( z ) 〉 ω .”

The stress-energy tensor operator is conserved, i.e., for every contravariant vector field X on M the stated conservation identity holds.

“The stress-energy tensor operator is conserved , i.e., for every contravariant vector field X on M it holds”

The tensor field t is built up, via standard tensor calculus, by employing the metric and the curvature tensors at z, m, ξ, V′(z) and their covariant derivatives at z.

“where the smooth symmetric tensor field t is independent from either the quantum state, λ and λ ′ , is conserved for V ′ ≡ 0 and it is built up, via standard tensor calculus, by employing the metric and the curvature tensors at z , m , ξ , V ′ ( z ) and their covariant derivatives at z .”

Remark (1): The introduced classically vanishing term :g_{X,Y}(x) φ(x) Pφ(x): is crucial to obtain the conservation of the stress-energy tensor operator using the analogous property of the point-splitting renormalized stress-energy tensor.

“(1) We have introduced the, classically vanishing, term : g X,Y ( x ) ϕ ( x ) Pϕ ( x ): . Its presence is crucial to obtain the conservation of the stress-energy tensor operator using the analogous property of the point-splitting renormalized stress-energy tensor as done in the proof of the theorem below.”

• AssumptionThe authors state this as an assumption: dim(M)=4 and V'≡0 in the Klein-Gordon operator.

For every h in C∞(M), the operator :h(x)φ(x)Pφ(x): does not depend on λ.

“For every h ∈ C ∞ ( M ) , : h ( x ) ϕ ( x ) Pϕ ( x ): does not depend on λ and turns out to be a smooth function.”

The solution is exact if one takes the limit Z=lim n→∞ Zn provided the limit exists.

“That solution is exact if one takes the limit Z = lim n →∞ Z n of the sequence provided the limit exist.”

If t ∈ E'_n(M,g), there is a well-defined operator-valued symmetric distribution D_n(M) ∋ f ↦ (t ⊙ Wω,n)(f) with values in Dω.

“Summarizing: if t ∈ E ′ n ( M, g ), n ∈ N , it is well-defined an operator-valued symmetric distribution D n ( M ) ∋ f ↦→ ( t ⊙ ˆ W ω,n ) ( f ), with values defined in the dense invariant domain D ω .”

• AssumptionThe authors state this as an assumption: Every considered operator is defined on Dω.

〈 ˆ φ ( z ) P ˆ φ ( z ) 〉 ω is proportional to the anomalous quantum correction to the trace of the stress-energy tensor.

“We conclude that 〈 ˆ ϕ ( z ) P ˆ ϕ ( z ) 〉 ω (i) does not depend on the scale λ , (ii) does not depend on ω and (iii) is proportional to the anomalous quantum correction to the trace of the stress-energy tensor.”

For every h in C∞(M), the operator :h(x)φ(x)Pφ(x): is a smooth function.

“For every h ∈ C ∞ ( M ) , : h ( x ) ϕ ( x ) Pϕ ( x ): does not depend on λ and turns out to be a smooth function.”

〈 ˆ φ ( z ) P ˆ φ ( z ) 〉 ω does not depend on ω.

“We conclude that 〈 ˆ ϕ ( z ) P ˆ ϕ ( z ) 〉 ω (i) does not depend on the scale λ , (ii) does not depend on ω and (iii) is proportional to the anomalous quantum correction to the trace of the stress-energy tensor.”

For any quasifree Hadamard state ν and any Hadamard state ω represented by Ψω in a GNS representation of ν, the point-split expectation values of the stress-energy tensor and of φ2 satisfy the stated identity for every pair of contravariant vector fields X,Y.

“Take a quasifree Hadamard state ν and let ω be any (not necessarily quasifree) Hadamard state represented by Ψ ω ∈ D ν ⊂ H ν in a GNS representation of ν . For every pair of contravariant vector fields X,Y , it holds”

The counterterms given by local Wick products of differentiated fields are independent from the arbitrary length scale and any quantum state.

“These terms are independent from the arbitrary length scale (and any quantum state) and they classically vanish on solutions of Klein-Gordon equation.”

The added terms :hφPφ(f): in the stress-energy tensor operator classically vanish.

“The added terms : hϕPϕ ( f ): in the form of the stress-energy tensor operator enjoy three remarkable properties. (1) They classically vanish, (2) they are written as local Wick products of field operators, (3) they are in a certain sense universal , i.e., they do not depend on the scale λ and belong to the commutant of the algebra W ( M, g ).”

G(1)ω, as well as G(+)ω, is a smooth function away from the subset of M × M made of pairs of points x, y such that either x = y or they are light-like related.

“It also implies that G (1) ω (as well as G (+) ω ) is a smooth function, ( x, y ) ↦→ G (1) ω ( x, y ) away from the subset of M × M made of the pairs of points x, y such that either x = y or they are light-like related.”

The added terms :hφPφ(f): in the stress-energy tensor operator are written as local Wick products of field operators.

“The added terms : hϕPϕ ( f ): in the form of the stress-energy tensor operator enjoy three remarkable properties. (1) They classically vanish, (2) they are written as local Wick products of field operators, (3) they are in a certain sense universal , i.e., they do not depend on the scale λ and belong to the commutant of the algebra W ( M, g ).”

〈 ˆ φ ( z ) P ˆ φ ( z ) 〉 ω does not depend on the scale λ.

“We conclude that 〈 ˆ ϕ ( z ) P ˆ ϕ ( z ) 〉 ω (i) does not depend on the scale λ , (ii) does not depend on ω and (iii) is proportional to the anomalous quantum correction to the trace of the stress-energy tensor.”

The counterterms given by local Wick products of differentiated fields classically vanish on solutions of the Klein-Gordon equation.

“These terms are independent from the arbitrary length scale (and any quantum state) and they classically vanish on solutions of Klein-Gordon equation.”

The averaged stress-energy tensor obtained by the point-splitting procedure coincides with that found by employing the local ζ-function approach whenever that technique can be implemented.

“The averaged stress-energy tensor obtained by the point-splitting procedure also coincides with that found by employing the local ζ -function approach whenever that technique can be implemented.”

Proposition 3.1: The sub *-algebra W_H(N,g) generated by W_H,n(t), t ∈ E'_n(N,g), n=0,1,..., coincides with W(N,g) itself and is naturally *-isomorphic to the sub *-algebra of W(M,g) whose elements are smeared by distributions with support in N^n.

“Proposition 3.1. Referring to the given definitions, the sub ∗ -algebra of W ( N, g ) , W H ( N, g ) , generated by W H,n ( t ) , t ∈ E ′ n ( N, g ) , n = 0 , 1 , . . . . (a) W H ( N, g ) coincides with W ( N, g ) it-self and (b) is naturally ∗ -isomorphic to the sub ∗ -algebra of W ( M, g ) whose elements are smeared by distributions with support in N n .”

• AssumptionThe authors state this as an assumption: dim(M)=4 and V'≡0 in the Klein-Gordon operator.

The classical stress-energy tensor T(η)μν(z) equals Tμν(z) whenever φ satisfies the field equation Pφ = 0.

“It is obvious that T ( η ) µν ( z ) = T µν ( z ) whenever ϕ satisfies the field equation Pϕ = 0.”

For ξ = ξD, if V ≡ 0 and m = 0, the action of the field φ is invariant under local conformal transformations.

“For ξ = ξ D , if V ≡ 0 and m = 0, the action of the field ϕ turns out to be invariant under local conformal transformations ( g ( x ) → λ ( x ) g ( x ), ϕ ( x ) → λ ( x ) 1 / 2 -D/ 4 ϕ ( x )) and the trace of T αβ ( x ) vanishes on field solutions by (5).”

The point-splitting renormalization defined above is in agreement with four of Wald's axioms.

“The point-splitting renormalization defined above turns out to be in agreement with four Wald's axioms.”

• cites [7] [7] R. M. Wald, 'Quantum field theory in curved space-time and black hole thermodynamics'. Chicago, USA: Univ. Pr. (1994) 205p . -Not matched

  Couldn’t be matched to a paper in Crossref, so it can’t be checked.

The tensor field t is independent from the quantum state, λ and λ′.

“where the smooth symmetric tensor field t is independent from either the quantum state, λ and λ ′ , is conserved for V ′ ≡ 0 and it is built up, via standard tensor calculus, by employing the metric and the curvature tensors at z , m , ξ , V ′ ( z ) and their covariant derivatives at z .”

The coefficients Uk(z,z) do not depend on either ω or the scale λ used in the definition of Zn.

“Notice that the coefficients U k ( z, z ) do not depend on either ω and the scale λ used in the definition of Z n .”

Aω(M,g) is a sub *-algebra of Wω(M,g) since φω(f) = :φ(f):ω for f ∈ D(M).

“A ω ( M, g ) turns out to be a sub ∗ -algebra of W ω ( M, g ) since one finds that ˆ ϕ ω ( f ) = : ˆ ϕ ( f ): ω for f ∈ D ( M ).”

• AssumptionThe authors state this as an assumption: Every considered operator is defined on Dω.

Lemma 3.1(a): The local Wick products on a causal domain N ⊂ M do not depend on the arbitrary terms ψ and {α_k} used in the definition of H (but may depend on the length scale λ).

“(a) The local Wick products on a causal domain N ⊂ M , do not depend on the arbitrary terms ψ and { α k } used in the definition of H (but may depend on the length scale λ ).”

• AssumptionThe authors state this as an assumption: dim(M)=4 and V'≡0 in the Klein-Gordon operator.

If 0 < λ′ ≠ λ, the difference between the stress-energy tensor operators at scales λ and λ′ equals the smooth, symmetric, conserved tensor field t introduced in (23).

“If 0 &lt; λ ′ = λ , with obvious notation, ̸”

• Also stated in: § 3 The stress-energy operator in terms of local Wick products. · p. 22 · plain

  “where the smooth, symmetric, conserved tensor field t is that introduced in (23) .”

The trace of the stress-energy tensor can be computed in terms of φ2(x) only for Pφ = 0.

“The trace of the stress-energy tensor can easily be computed in terms of ϕ 2 ( x ) only. In fact, for Pϕ = 0, one finds”

The averaged stress-energy tensor and field fluctuation satisfy the analogue of the conservation equation for all spacetimes if and only if η = ηD = D[2(D+2)]^{-1}.

“Moreover, if (and only if) η = η D def = D [2( D +2)] -1 , they satisfy the analogue of (4) for all spacetimes”

The conserved quantity is a term corresponding to the classical stress-energy tensor added to a further term depending on the functional measure.

“The conserved quantity is a term corresponding to the classical stress-energy tensor added to a further term depending on the functional measure”

• AssumptionThe authors state this as an assumption: The measure Dgφ in general depends on the metric.

The need of adding a term to the classical stress-energy tensor to fulfill the conservation requirement can be heuristically explained by the functional measure depending on the metric.

“The relevant point is that the measure D g ϕ in general must be supposed to depend on the metric [17, 6, 18].”

• cites [17] [17] S. W. Hawking, 'Zeta Function Regularization Of Path Integrals In Curved SpaceTime'. Commun. Math. Phys. 55 , 133 (1977). - sourceNot checked — no full text of the source
• cites [6] [6] N. D. Birrell and P. C. Davies, 'Quantum Fields In Curved Space'. Cambridge, UK: Univ. Pr. (1982) 340p . - sourceNot checked — no full text of the source
• cites [18] [18] K. Fujikawa, 'Comment On Chiral And Conformal Anomalies'. Phys. Rev. Lett. 44 , 1733 (1980). - sourceNot checked — no full text of the source

• AssumptionThe authors state this as an assumption: The measure Dgφ in general depends on the metric.
• AssumptionThe authors state this as an assumption: There is some functional of the metric corresponding to the one-loop effective action.

A definition of stress-energy tensor operator in curved spacetime is possible in terms of local Wick products of field operators only by adding suitable terms to the classical form of the stress energy tensor.

“We have shown that a definition of stress-energy tensor operator in curved spacetime is possible in terms of local Wick products of field operators only by adding suitable terms to the classical form of the stress energy tensor.”

The found term ηD gμν(z) 〈 φ(z) Pφ(z) 〉 ω added to the classical stress-energy tensor should be related to the second term in the brackets above.

“Therefore the found term η D g µν ( z ) 〈 ϕ ( z ) Pϕ ( z ) 〉 ω added to the classical stress-energy tensor should be related to the second term in the brackets above.”

• AssumptionThe authors state this as an assumption: The measure Dgφ in general depends on the metric.