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Finite Temperature Field Theory Yuri V. Gusev Lebedev Research Center in Physics, Moscow CCGRRA, July 6, 2016 Yuri V. Gusev (Lebedev Research Center in Physics,Finite Moscow) Temperature Field Theory CCGRRA, July 6, 2016 1 / 17 Paper Finite temperature quantum field theory in the heat kernel method, Russian Journal of Mathematical Physics 22 (1), 9-19 (2015). Available at MPI publications (Albert Einstein Institute) aei.mpg.de. Linked by ADS Harvard, inSPIRE HEP. Different from Yu.V. Gusev and A.I. Zelnikov, Phys. Rev. D 59, 024002(12) (1998). Yuri V. Gusev (Lebedev Research Center in Physics,Finite Moscow) Temperature Field Theory CCGRRA, July 6, 2016 2 / 17 Laplace operator A field model is defined the differential Laplace operator F (∇) of the generic type 1 F̂ (∇) = 21̂ + P̂ − R 1̂, 6 The Laplace-Beltrami operator 2 ≡ g µν ∇µ ∇ν ∇µ is a covariant derivative with respect to the metric and gauge field. Potential P̂ is a local function. Yuri V. Gusev (Lebedev Research Center in Physics,Finite Moscow) Temperature Field Theory CCGRRA, July 6, 2016 3 / 17 The kernel of the evolution (heat) equation Heat kernel: (Euclidean) geometrical approach to the field theory via the evolution equation over the proper time s ∂ K̂ (s|x, y ) = F̂ (∇x )K̂ (s|x, y ) ∂s Its funcitional trace: the coincident points K̂ (s|x, x), the matrix trace, and the spacetime integral Z TrK (s) ≡ d D xtrK̂ (s|x, x) Arbitrary integer dimension D ≥ 3. Yuri V. Gusev (Lebedev Research Center in Physics,Finite Moscow) Temperature Field Theory CCGRRA, July 6, 2016 4 / 17 The effective action The definition of the full efective action via the evolution kernel, Z ∞ ds −W ≡ A TrK (s|x, x). s 0 Geometric functional TrK ⇒ the effective action W mathematics ⇒ physics The calibration constant A is fixed by experiment. Yuri V. Gusev (Lebedev Research Center in Physics,Finite Moscow) Temperature Field Theory CCGRRA, July 6, 2016 5 / 17 Topology Any condensed matter system is confined to the compact domain MD of the space RD , with the boundary B D−1 . The effects of boundary and edges of a spatial domain should always be accounted for. Experiments show the size and boundary effects of a condensed matter system can be large and even leading. Euclidean spacetime D = d + 1, with the closed Euclidean time S1 . Computation of the heat kernel with non-trivial topology. No thermodynamic temperature yet. Yuri V. Gusev (Lebedev Research Center in Physics,Finite Moscow) Temperature Field Theory CCGRRA, July 6, 2016 6 / 17 Spatial domain The volume of the D-dimensional domain is Z V= dD xg 1/2 (x), M The area of its (D − 1)-dimensional smooth boundary is Z S= dD−1 x ḡ 1/2 (x). B where ḡ is the metric determinant of the boundary manifold. Yuri V. Gusev (Lebedev Research Center in Physics,Finite Moscow) Temperature Field Theory CCGRRA, July 6, 2016 7 / 17 Evolution kernel The fundamental solution of the evolution equation, σ(x, x 0 ) 1 1/2 0 â0 (x, x 0 ), K̂ (s|x, x 0 ) = D (x, x ) exp − 2s (4πs)D/2 σ(x, x 0 ) the Ruse-Synge world function, â0 (x, x 0 ) is the parallel transport operator, D(x, x 0 ) is the van Vleck-Morette determinant. The kernel of the evolution equation in the compact domain M with the boundary B, TrK (s) = 1 1 V tr 1̂ + S tr1̂ + O[<]. D/2 (D−1)/2 (4πs) (4πs) This expression is valid at arbitrary proper time s. It is covariant even though the curvatures and field strengths do no appear explicitly. Yuri V. Gusev (Lebedev Research Center in Physics,Finite Moscow) Temperature Field Theory CCGRRA, July 6, 2016 8 / 17 Thermal field theory Spacetime with the ultrastatic metric ds 2 = gµν dx µ dx ν = dτ 2 + ḡij (x)dx i dx j Metric ḡij (x) depends only on the spatial coordinates x. Always true for thermostatics. The world function in the Euclidean time coordinate τ is trivial, σ(τ, τ 0 ) = (τ − τ 0 )2 /2. The volume of the closed 1D space is β, integer n counts the windings. Evolution kernel is factorized, Z ∞ 2 2 X β − β 4sn e TrK (β|s) = dd x tr K (d) (s|x, x), d ≥ 2. (4πs)1/2 n=1 No ’zero mode’ (no theta functions). n = 0 does not satisfy the evolution equation. At least one loop to produce the curvatures. Yuri V. Gusev (Lebedev Research Center in Physics,Finite Moscow) Temperature Field Theory CCGRRA, July 6, 2016 9 / 17 Free energy The definition of the free energy, Z −F (β) ≡ A ∞ 0 ds TrK (β|s) s by replacement y = β 2 /4s the integral is, Z 0 ∞ dy y a−1 ∞ X 2 e−yn = ζ(2a) Γ(a). n=1 ζ - Riemann zeta function, Γ - gamma function. Yuri V. Gusev (Lebedev Research Center in Physics,Finite Moscow) Temperature Field Theory CCGRRA, July 6, 2016 10 / 17 Free energy at arbitrary temperature For three spatial dimensions (standard condensed matter), −F (β) = 1 π 2 (3) 1 ζ(3) (3) V tr1̂ + 2 S tr1̂ + O[<], d = 3. β 3 90 β 2π The free energy in (2+1) dimensions, −F (β) = 1 ζ(3) (2) 1 π (2) V tr1̂ + S tr1̂ + O[<], d = 2. 2 β 2π β 6 The free energy (’the finite temperature effective action’) is finite. Other contributions arise due to the non-smooth bulk/boundaries (’curvatures’). Yuri V. Gusev (Lebedev Research Center in Physics,Finite Moscow) Temperature Field Theory CCGRRA, July 6, 2016 11 / 17 Planck’s inverse temperature New thermal variable β scaled as meter . There is nor temperature, neither (Euclidean) time. β= ~v BTkB kB - the Boltzmann’s constant, ~ - the Planck’s constant. B is the second calibration constant. The characteristic velocity v : electronic phenomena - the speed of light c, elastic (acoustic) phenomena - the speed of sound va (in condensed matter, not in gas). Yuri V. Gusev (Lebedev Research Center in Physics,Finite Moscow) Temperature Field Theory CCGRRA, July 6, 2016 12 / 17 ’High temperature’ asymptotic Free energy is defined by two geometrical characteristics of a system, volume and boundary’s area. The boundary, 1/β 2 , term may become larger than the volume, 1/β 3 , one for certain physical conditions. It is meaningless to talk about the high or low temperature limit in terms of T , i.e. without knowing the ’size’ of a system. Effective size r of a domain as the ratio of its volume to its boundary’s area, V r≡ S e.g. the effective size of a sphere of diameter L is r = L/6. An expansion can be done only in a small dimensionless parameter. The ’high temperature’ limit is the asymptotic, β/r 1 The restriction on ”how large a body or a cavity should be compared to its thermodynamic temperature” for the ’high temperature’ limit. Yuri V. Gusev (Lebedev Research Center in Physics,Finite Moscow) Temperature Field Theory CCGRRA, July 6, 2016 13 / 17 ’Low temperature’ asymptotic Free energy (d = 3) rearrangement, 90ζ(3) β F (β) = FV (β) 1 + 2π 3 r The ’low temperature’ regime, κ ∝ 1.75 ~v > rT , kB the boundary term is larger than the volume term. ’Thermal wavelength’ in CMT (v = c) β ≈ 3.8 µm. For acoustic phenomena, va ≈ 103 m, κ ≈ ·10−9 K m : the size effects appear at the 106 smaller values of rT . Yuri V. Gusev (Lebedev Research Center in Physics,Finite Moscow) Temperature Field Theory CCGRRA, July 6, 2016 14 / 17 No zero temperature True low temperature asymptotic β r is in the remainder O[<]. May become dominant. Specific of the physical system’s properties (material, surface curvatures, etc). No universal low temperature asymptotic for free energy and its derivatives exist. Variety of condensed matter effects towards ’absolute zero’ T . Can be described by quasiparticles (polaron, exciton, polariton, plasmon, etc). T = 0 is forbidden topologically: no change from the closed manifold S1 to open R1 . Yuri V. Gusev (Lebedev Research Center in Physics,Finite Moscow) Temperature Field Theory CCGRRA, July 6, 2016 15 / 17 Summary Planck’s inverse temperature β is the proper thermal variable. Free energy is a dimensionless phenomenological functional. It is to generate the effective equations. No (UV or IR) divergences, no renormalization. No chemical potentials in finite temperature field theory. Similar to thermodynamics. The field formalism for condensed matter physics. Yuri V. Gusev (Lebedev Research Center in Physics,Finite Moscow) Temperature Field Theory CCGRRA, July 6, 2016 16 / 17 Specific heat theory Yu.V. Gusev, The field theory of specific heat. Russ. J. Math. Phys. 23 (1), 56-76 (2016). Dimensionless thermodynamical variable with the lattice constant a, ~v α= BTkB a The molar specific heat for the cubic lattice, l is the number of atoms per unit cell, kB NA CM (α) = A Θ(α), l is defined by the universal thermal sum, Θ(α). The ’low temperature’ asymptotics, 1 Θ(α) ∝ 4 , α → ∞. α The fourth order in temperature is well confirmed experimentally. Yuri V. Gusev (Lebedev Research Center in Physics,Finite Moscow) Temperature Field Theory CCGRRA, July 6, 2016 17 / 17