Download Finite Temperature Field Theory

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