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Variational Derivatives and Integrals

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Variational Derivatives and Integrals Appendix A Variational Derivatives and Integrals Definitions and useful properties of functionals, variational derivatives and functional integrals are collected here, and illustrated for the well known example of a classical scalar field. A.1 Functional Derivatives A functional, F[ f ], is a mapping from a set of functions on configuration space to the real or complex numbers. We will denote the value of f at x by fx . The functional derivative of F[ f ] is defined via the variation of F with respect to f , i.e., δF δF := F[ f + δ f ]−F[ f ]= dx δfx (A.1) δ fx for arbitrary infinitesimal variations f → f + δ f . Thus the functional derivative is a field density, δF/δ f , having the value δF/δ fx at position x. Note that this definition is analogous to the definition of the partial derivative of a function g(x) via g(x + ε) − g(x) = ε ·∇g(x) for arbitrary infinitesimal variations x → x + ε. It follows directly from Eq. (A.1) that the functional derivative satisfies product and chain rules analogous to ordinary differentiation. The choice F[ f ]= fx in Eq. (A.1) yields δ fx /δ fx = δ(x − x ). (A.2) Moreover, if the field depends on some parameter, t say, then choosing δ fx = fx (t + δt) − fx (t) in Eq. (A.1) yields © Springer International Publishing Switzerland 2016 273 M.J.W. Hall and M. Reginatto, Ensembles on Configuration Space, Fundamental Theories of Physics 184, DOI 10.1007/978-3-319-34166-8 274 Appendix A: Variational Derivatives and Integrals dF ∂ F δF ∂ f = + dx x (A.3) dt ∂t δ fx ∂t for the rate of change of F with respect to t. As another useful example, for a functional of the form F = dx g(x, f, ∇ f ) one has δF = dx [g(x, f + δ f, ∇ f +∇(δ f )) − g(x, f, ∇ f )] ∂g ∂g = dx δ f + ∇(δ f ) ∂ f ∂∇ f ∂g ∂g = dx −∇· δ f, ∂ f ∂∇ f where integration by parts has been used in the last line, assuming that the variations and/or the derivatives of g vanish at infinity. Hence from Eq. (A.1) one has δF/δ f = ∂g/∂ f −∇·∂g/∂(∇ f ) (A.4) for this case. This formula is easily extended when g also depends on higher deriv- atives of f . A.2 Functional Integrals Functional integrals correspond to integration of functionals over the space of func- tions (or equivalence classes thereof). We will consider here a measure Df on this vector space which is translation invariant, i.e., Df ≡ Df for any translation f = f +h (which follows immediately, for example, from approaches to functional integration based on discretising the space of functions). In particular, this property implies the useful result δF Df = 0for DfF[ f ] < ∞. (A.5) δ f Equation(A.5) follows by noting that the finiteness condition and translation invari- ance imply δF 0 = Df (F[ f + δ f ]−F[ f ]) = dx δfx Df (A.6) δ fx for arbitrary infinitesimal translations, where we use fx to denote the value of f at x. Thus, for example, if F[ f ] has a finite expectation value with respect to some probability density functional P[ f ], then Eq. (A.5) yields the “integration by parts” formula Appendix A: Variational Derivatives and Integrals 275 δF δP DfP =− Df F. (A.7) δ f δ f Moreover, from Eq. (A.5) the total probability, DfP, is conserved for any proba- bility flow satisfying a continuity equation of the form ∂ P δ + dx [PVx ] = 0, (A.8) ∂t δ fx providing that the average flow rate, Vx , is finite. Finally, consider a functional integral of the form I [F]= Df ξ(F,δF/δ f ), (A.9) where ξ denotes any function of some functional F and its functional derivative. Variation of I [F] with respect to F then gives, to first order, ΔI = I [F + ΔF]−I [F] ∂ξ ∂ξ δ(ΔF) = Df ΔF + dx ∂ F ∂(δF/δ f ) δ f x x ∂ξ δ ∂ξ = Df − dx ΔF ∂ F δ f ∂(δF/δ f ) x x δ ∂ξ + dx Df ΔF . (A.10) δ fx ∂(δF/δ fx ) Assuming that the functional integral of the expression in curly brackets in the last term is finite, this term vanishes from Eq. (A.5), yielding the result ΔI ΔI = Df ΔF (A.11) ΔF analogous to Eq. (A.1), where the variational derivative ΔI/ΔF is defined by ΔI ∂ξ δ ∂ξ := − dx . (A.12) ΔF ∂ F δ fx ∂(δF/δ fx ) 276 Appendix A: Variational Derivatives and Integrals A.3 Example: Ensemble Hamiltonian for a Classical Scalar Field To illustrate the application of the above concepts, we consider a classical scalar field, its corresponding ensemble Hamiltonian, and the equations of motion that follow from this ensemble Hamiltonian. Field theories present well known mathematical and conceptual difficulties. As a consequence, the equations are formal in nature and it is necessary to examine each individual field theory to establish the validity of the ensemble formalism when applied to particular cases. This is done whenever particular field theories are dis- cussed in the monograph. In the case of a field theory, the configuration is described by some field φ(x) (which may comprise multi-component fields that that carry a set of indices, which we omit here to simplify the notation). The probability density P[φ] is a functional, defined over the space of fields. We can consider a Hamiltonian description of the time evolution of the field, and introduce an auxiliary quantity S[φ] that is canonically conjugate to P[φ], as discussed in Chap.5. As an example, consider a classical scalar field φ. In the classical Hamilton–Jacobi formulation reviewed in Chap.5, the equation for φ is given by ∂ S 1 δS 2 + dx + |∇φ|2 + V (φ) = 0. (A.13) ∂t 2 δφ The ensemble Hamiltonian H is the functional integral corresponding to the mean energy of the field, 1 δS 2 H [P, S]:= Df dxP + |∇φ|2 + V (φ) . (A.14) 2 δφ The Hamiltonian equations of motion for the dynamical variables P and S, ∂ P ΔH ∂ S ΔH = , =− , (A.15) ∂t ΔS ∂t ΔP reduce via Eq. (A.12) to the Hamilton–Jacobi equation, Eq. (A.13), as required, and to the continuity equation ∂ P δ δS + dx P = 0. (A.16) ∂t δφ δφ Comparison with Eq. (A.8) shows that the local rate of change of the field is V = δS/δφ. Index A Complex structure, 122 Action principle, 4 Condon, E.U., 214 Aharonov, Y., 32 Configuration space, 4 Aleksandrov, I.V., 193 discrete, 7 Constraints, 40 Copenhagen interpretation, 177 B Bell, J.S., 11, 141 Bergmann, P.G., 107 D Bhattacharyya, A., 117 Decoherence Bialynicki-Birula, I., 96 effective, 211, 214, 216 Black hole ineffective, 211 ensembles on configuration space, 232 Demers, J.-G-, 255, 261, 264 Hamilton-Jacobi formulation, 232 DeWitt measure, 227 Lemaître metric, 234, 237, 239, 240 DeWitt supermetric, 225, 227, 248 Bohm, D., 141 DeWitt, B.S., 244–246 Bohr, N., 174 Diosi, L., 192 Butterfield, J., 244 Dirac, P.A.M., 58 Doebner, H.-D., 128 Dressel, J., 32 C Dyson, F., 162, 244 Callender, C., 245 Campbell, L.L., 119, 120 Caro, J., 193 E Casimir functions, 144, 154 Einstein-Hamilton-Jacobi equation, see Casimir operator, 144, 155 Hamilton-Jacobi formulation, gravi- Cencov, N.N., 119, 120 tation CGHS black hole, 255 Electromagnetic field, 102 action, 255 Lorentz gauge, 102 collapsing space-time, 261 exact uncertainty quantization, 103 ensemble Hamiltonian, 257 radiation gauge, 103 with a quantized scalar field, 257 exact uncertainty quantization, 104 ensemble Hamiltonian, 257 Elze, H.-T., 192 gravitational time, 266, 269 Ensemble Hamiltonian Hawking radiation, 264, 269 as generator of time translations, 23 perturbative expansion, 258, 261 classical particle, 5 Classicality, 165, 207–209, 216 conservation of probability, 10 © Springer International Publishing Switzerland 2016 277 M.J.W. Hall and M. Reginatto, Ensembles on Configuration Space, Fundamental Theories of Physics 184, DOI 10.1007/978-3-319-34166-8 278 Index discrete classical system, 8 H discrete quantum system, 9 Hamilton-Jacobi formulation gravitation, 228, 247 fields, 98, 107, 109 gravitation and quantized scalar field, gravitation, 224 248 rate equation, 229 homogeneity, 13 reconstruction problem, 226, 235 positivity of probability, 12 particles, 5 quantum particle, 6 spherically symmetric gravity, 230 spherically symmetric gravity, 231, 249 rate equation, 231 Ensembles on configuration space, 4 Hannah, E., 244–246 distinguishable, 69 Hawking eigenensemble, 30 radiation, 264 eigenvalue, 30 Hawking, S.W., 107 entangled, 47 Heisenberg, W., 86, 87, 174 fields Huggett, N., 245 classical, 98, 109 Hybrid cosmological model, 250 quantum, 99 midisuperspace, 252 geometry, 124 minisuperspace, 250 complex coordinates, 127 Hybrid ensemble, 165 hybrid, see hybrid ensemble centre-of-mass motion, 173 independent, 44, 165 classical particle and quantum spin, 166 interacting, 166 classical-quantum scattering, 177 joint, 43 conditional density operator, 170 noninteracting, 45, 166 ontology, 20 conditional wave function, 170 stationary, 29, 69 configuration separability, 169, 199 weak value, 31 dynamical bracket, 193 Entanglement, 46 Ehrenfest relations, 196 Eppley, K., 244–246 Galilean invariance, 174 Extrinsic curvature, 235, 237 harmonic oscillator, 180 coherent ensemble, 182, 183 Gaussian ensemble, 181 F interacting particles, 166 Feynman, R.P., 244, 246 ensemble Hamiltonian, 166 Fisher information matrix, 118 measurement interactions, 175 Fisher length, 88, 96 measurement model, 209 Fisher–Rao metric, 118 measurement of position, 175 measurement of spin, 176 measuring device, 211 G mixture Galilean group, 35 improper, 211, 216 Gamow, G., 214 proper, 216 Geilker, C.D., 244, 246 reduced, 169, 170, 211 Gerlach, U.H., 226, 229 momentum separability, 199 Giulini, D., 225 no-go theorems, 195 Goldin, G.A., 128 observables, 168 Goyal, P., 133 oscillator benchmark, 196 Gravitational field, 104 relative motion, 173 classical ensemble, 105, 229 stationary, 171, 172 intrinsic concept of time, 229 strong separability, 201, 202, 214,
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