By applying Schwinger's variational principle to the Einstein-Cartan action
for the gravitational field, we derive quantum commutation relations between
the metric and torsion tensors.Comment: 3 page

Poplawski, Nikodem

Physical Review D

English

arXiv

By applying Schwinger’s variational principle to the Einstein-Cartan action for the gravitational field, we derive quantum commutation relations between the metric and torsion tensors

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University of New HavenDigital Commons @ New HavenPhysics Faculty Publications Physics1-15-2014Schwinger’s Variational Principle in Einstein-CartanGravityNikodem PoplawskiUniversity of New Haven, NPoplawski@newhaven.eduFollow this and additional works at: http://digitalcommons.newhaven.edu/physics-facpubsPart of the Physics CommonsCommentsThis is the author's submitted version of the article that appeared in Physical Review D, available at:http://dx.doi.org/10.1103/PhysRevD.89.027501. Originally posted at arXiv:1310.8014 [gr-qc]Publisher CitationPoplawski, Nikodem (2014). Schwinger’s variational principle in Einstein-Cartan gravity. Physical Review D 89(2), January 15, 2014,027501.arXiv:1310.8014v1  [gr-qc]  30 Oct 2013Schwinger’s principle in Einstein-Cartan gravityNikodem Pop lawskiDepartment of Physics, University of New Haven,300 Boston Post Road, West Haven, CT 06516, USA∗By applying Schwinger’s variational principle to the Einstein-Cartan action for the gravitationalﬁeld, we derive quantum commutation relations between the metric and torsion tensors.The ﬁeld equations of a classical theory are obtained in Hamilton’s principle of stationary action by varying theaction with respect to every variable φ and equating such a variation to zero for an arbitrary variation δφ. Equivalently,we calculate the variation of the Lagrangian density in the action integral and perform integration by parts. Then weapply the Gauß-Stokes theorem and assume δφ = 0 on the boundary of the integration domain. Finally, we equateto zero the coeﬃcients of δφ inside the integration domain [1].In a quantum theory, Hamilton’s principle is generalized to Schwinger’s variational principle, according to whichthe variation of the transition amplitude between an initial state |αi〉 and a ﬁnal state |αf 〉 is i/~ times the matrixelement connecting the two states of the variation δI of the action integral I [2, 3]:δ〈αf |αi〉 = i~〈αf |δI|αi〉. (1)If Schwinger’s principle is applied to variations which vanish on the boundary of the integration domain, the left-handside of (1) is equal to zero. Since the states |αi〉 and |αf 〉 are arbitrary, the operator δI vanishes, giving the principleof stationary action in an operator form. If Schwinger’s principle is applied to variations which vanish only for theinitial state, the eﬀect of the variation in the Schro¨dinger picture is to apply a unitary operator U = 1 + (i/~)δI tothe adjoint ﬁnal state, which is equivalent to applying the Hermitian conjugate U † to the ﬁnal state, |αf 〉 → U †|αf 〉.In the Heisenberg picture, the eﬀect of this variation is to apply to any operator O acting on the ﬁnal state a unitarytransformation O → UOU−1 = O− (i/~)[O, δI], where square brackets denote a commutator. Schwinger’s variationalprinciple is thus equivalent to [2–4]δO = − i~[O, δI]. (2)For a particle in quantum mechanics, we substitute a coordinate operator qi for O and use the formula for theaction as a function of the coordinates and time at the ﬁnal state, δI = δ∫Ldt =∑pjδqj − Hδt, where L is theLagrangian, pj = ∂L/∂q˙j denotes the generalized momentum corresponding to the generalized coordinate qj , andH =∑pj q˙j −L is the Hamiltonian [5]. The term δI is the variation of the action at the boundary of the integrationdomain (over time). If we consider a variation in which only the coordinates qi are changed, then δI =∑pjδqjand Schwinger’s principle (2) gives δqi = −(i/~)[qi,∑pjδqj ], which is satisﬁed if [qi, qj ] = 0 and [qi, pj ] = i~δij . Byadding to the action I a quantity −∑pjqj , we bring its variation to δI = −∑ qjδpj . Schwinger’s principle (2) givesδpi = (i/~)[pi,∑qjδpj], which is satisﬁed if [pi, pj ] = 0 and [pi, qj ] = −i~δij. Schwinger’s principle therefore gives thecanonical commutation relations of quantum mechanics [2–4].For the electromagnetic ﬁeld, the Lagrangian density is equal to L = (1/4)∫FµνFµν√−g, where Fµν = ∂µAν−∂νAµis the electromagnetic ﬁeld tensor, Aµ is the electromagnetic potential, and g = det(gµν) is the determinant of themetric tensor gµν [6, 7]. The action is thus equal to I = −(1/4c)∫FµνFµν√−gdΩ, where dΩ is an element of four-volume. The variation of the actions is given by δI = −(1/c) ∫ Fµν∂µδAν√−gdΩ = −(1/c) ∫ ∂µ(Fµν√−gδAν)dΩ +(1/c)∫∂µ(Fµν√−g)∂AνdΩ = −(1/c)∮Fµν√−gδAνdfµ+(1/c)∫∂µ(Fµν√−g)∂AνdΩ, where dfµ is an element of theclosed hypersurface surrounding the integration four-volume. We can consider a volume hypersurface dfµ = δ0µdV ,where dV is an element of volume. We also use g = −g00γ, where γ = det(γαβ) is the determinant of the spatial metrictensor γαβ = −gαβ+g0αg0β/g00 [6, 7]. The hypersurface term which should be used in Schwinger’s principle (2) is thusδI = −(1/c) ∫ √g00F 0βδAβ√γdV . This principle gives δAα(x, t) = (i/~c)[Aα(x, t), ∫ (√g00F 0βδAβ√γ)(x′, t)dV ′],which is satisﬁed if [Aα(x, t), gµν(x′, t)] = [Aα(x, t), Aβ(x′, t)] = 0 and [Aα(x, t), (√g00F0β√γ)(x′, t)] = −i~cδβαδ(x −x′). Deﬁning the spatial vector of the electric ﬁeld, Eα = −√g00F 0α (such a vector appears in the Maxwell equationsin curved space) [6], gives [Aα(x, t), Eβ(x′, t)] = i~cδβαδ(x−x′)/√γ(x), which generalizes to curved space the canonicalcommutation relations of electrodynamics [2, 4].∗Electronic address: NPoplawski@newhaven.edu2For the gravitational ﬁeld, we use the Einstein-Cartan theory of gravity, which naturally extends the general theoryof relativity to matter with intrinsic angular momentum (spin) [7, 8]. In this theory, the aﬃne connection Γ ρµ νis not constrained to be symmetric. Its antisymmetric part, the torsion tensor Sρµν = Γρ[µν], is a variable in theprinciple of stationary action. Square brackets denote antisymmetrization. Regarding the metric and torsion tensorsas independent variables gives the correct generalization of the conservation law for the total (orbital plus intrinsic)angular momentum to the presence of the gravitational ﬁeld [7, 8]. Including torsion also removes the curvaturesingularity at the big bang and replaces it with a nonsingular bounce [9], solves the ﬂatness and horizon problems incosmology [10], and requires fermions to be spatially extended, which may provide an ultraviolet cutoﬀ in quantumﬁeld theory [11].We use the notations of [7]. The metricity condition gµν;ρ, where semicolon denotes the covariant derivative withrespect to Γ ρµ ν , gives Γρµ ν = { ρµ ν}+Cρµν , where { ρµ ν} = (1/2)gρσ(gµσ,ν + gνσ,µ− gµν,σ) are the Christoﬀel symbols ofthe metric, gµν is the contravariant metric tensor, and Cρµν = Sρµν+2Sρ(µν) is the contortion tensor. Round bracketsdenote symmetrization. The curvature tensor Rλρµν = Γλρ ν,µ − Γ λρ µ,ν + Γ σρ νΓ λσ µ − Γ σρµΓ λσ ν , where comma denotesa partial derivative over coordinates, can be thus decomposed as Rλρµν = Pλρµν + Cλρ ν:µ − Cλρ µ:ν + Cσρ νCλσ µ −Cσρ µCλσ ν , where Pλρµν is the curvature tensor constructed from the Christoﬀel symbols (the Riemann tensor) andcolon denotes the covariant derivative with respect to { ρµ ν} [7, 12]. The Lagrangian density for the gravitational ﬁeldin the Einstein-Cartan theory of gravity is the same as in the general theory relativity, L = −(1/2κ)Rµνgµν√−g,where Rµν = Rρµρν is the Ricci tensor. The Einstein-Cartan action is thus given by [7]I = − 12κc∫Rµνgµν√−gdΩ = − 12κc∫gµν(Pµν − 2Cρµρ:ν − CρµρCσνσ + CρµσCσνρ)√−gdΩ= − 12κc∫gµν(Pµν − CρµρCσνσ + CρµσCσνρ)√−gdΩ+ 1κc∫(gµνCρµρ√−g),νdΩ. (3)The last term in (3) can be written as a hypersurface integral,IS =2κc∫gµνSµdfν , (4)where gµν =√−ggµν is the contravariant metric tensor density and Sµ = Sνµν is the torsion vector.Let us consider the variation of the action with respect to the torsion tensor. Such a variation at the boundary ofthe integration domain is thus equal toδIS =2κc∫gµνδSµdfν =2κc∫gµ0δSµdV. (5)Substituting this variation as δI into Schwinger’s principle (2) for O = Sµ givesδSµ(x, t) = − 2i~κc[Sµ(x, t),∫(gν0δSν)(x′, t)dV ′], (6)which is satisﬁed if[Sµ(x, t), Sν(x′, t)] = 0 (7)and[Sµ(x, t), gν0(x′, t)] =i2~κcδνµδ(x− x′) = 4piil2Pδνµδ(x − x′), (8)where lP is the Planck length.The quantum commutation relation (8) between the metric and torsion indicates that: 1) the components gµ0 ofthe metric tensor cannot be zero, and 2) the torsion vector cannot be zero. Accordingly, exact spherically and axiallysymmetric gravitational ﬁelds do not exist in the quantum theory. In the classical Einstein-Cartan theory of gravitycoupled to spinors, the only irreducible components of the torsion tensor that do not vanish are the components of thecompletely antisymmetric torsion tensor, so the torsion vector vanishes. The relation (8) therefore endows spacetime,even in vacuum, with intrinsic torsion.[1] E. Schro¨dinger, Space-Time Structure (Cambridge Univ. Press, 1954).3[2] J. Schwinger, Phys. Rev. 82, 914 (1951); Phys. Rev. 91, 713 (1953).[3] W. Yourgrau and S. Mandelstam, Variational Principles in Dynamics and Quantum Theory (Dover, 1979).[4] S. J. Chang, Introduction to Quantum Field Theory (World Scientiﬁc, 1990).[5] L. D. Landau and E. M. Lifshitz, Mechanics (Pergamon, 1976).[6] L. D. Landau and E. M. Lifshitz, The Classical Theory of Fields (Pergamon, 1975).[7] N. J. Pop lawski, arXiv:0911.0334.[8] T. W. B. Kibble, J. Math. Phys. 2, 212 (1961); D. W. Sciama, in Recent Developments in General Relativity (Pergamon,1962), p. 415; Rev. Mod. Phys. 36, 463 (1964); 36, 1103 (1964); F. W. Hehl and B. K. Datta, J. Math. Phys. 12, 1334(1971); F. W. Hehl, P. von der Heyde, G. D. Kerlick, and J. M. Nester, Rev. Mod. Phys. 48, 393 (1976); E. A. Lord,Tensors, Relativity and Cosmology (McGraw-Hill, 1976); V. de Sabbata and M. Gasperini, Introduction to Gravitation(World Scientiﬁc, 1985); V. de Sabbata and C. Sivaram, Spin and Torsion in Gravitation (World Scientiﬁc, 1994); N.Pop lawski, arXiv:1304.0047.[9] N. Pop lawski, Phys. Rev. D 85, 107502 (2012).[10] N. J. Pop lawski, Phys. Lett. B 694, 181 (2010); 701, 672 (2011).[11] N. J. Pop lawski, Phys. Lett. B 690, 73 (2010).[12] J. A. Schouten, Ricci Calculus (Springer-Verlag, 1954).

Schwinger's principle in Einstein-Cartan gravity

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