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有限温度量子场论-第二章 虚时路径积分(2.1 核心公式 2.2简谐振子模型)

第二章正式进入场论的计算。
平衡态巨正则系综可以表示成一个路径泛函。可以由场论方法非常自然地给出统计关系。

Chapter 2 Imaginary-Time Path Integral

2.1 The core formula in quantum statistic physics

The grand partition function is:

Z=Re(ERi=1MμiNi)/T

And for a quantum system with Halmitonian , since the number operators commute with , the (written in energy reprezentation)is:

Z=RR|e(H^i=1MμiN^i)/T|R

For interacting system, it is very hard to calculate all eigen states and eigen energy. Generally, we can calculate the trace under any representation:

(3)Z=Tr[eβ(H^i=1MμiN^i)]

Then we can define the density matrix:

ρ=[eβ(H^i=1MμiN^i)]

And for any operator , the thermal average value is:

(5)O^=1ZTr(O^ρ)

The equation (3) and (5) are the core formula in quantum statistic physics.

2.2 Thermodynamics of a harmonic oscillator

Harmonic oscillator is a simple but important model in quantum mechanics. It has a simple Halmitonian and energy levels:

H^=P^22m+12mω2X^2,En=ω(n+12)

QM = (0+1)-dimensional QFT: .

The partition function can be easily calculated. First we use the imaginary-time path integral to get the partition func. Then we can compare it to the result from the direct calculation of (3).

In energy representation, can be easily calculated:

Z=Tr(eβH^)=n=0n|eβH^|n=n=0eβω(n+12)=eβω/21eβω=12sinh(12βω)

The thermal avg of energy:

E=1ZTr(H^eβH^)=lnZβ=ω(12+1eβω1)=12ω+ωeβω1

At low , , .

At high , , .

Path integal: using x-basis.

Z=dxx|eβH^|x

Compare to the propergation amplitude in path integal:

A(t,x;t,x)=x|eiH^(tt)/|x

With imaginary time interval we can write the partition func as the form of path integal (divide the time interval to parts: ):

Z=dxx|i=1NeϵH^|x=dxx|eϵH^eϵH^eϵH^|x.

Then we do the trick of inserting orthogonal complete basis:

dx|xx|=1^,xx=δ(xx)dp|pp|=1^,pp=δ(pp)xp=12πeipx

For each time interval , we insert a basis and a basis:

x|eΔτH^eΔτH^eΔτH^|x=i=1NdxidpixN+1pNpN|eΔτH^|xNxNpN1pN1|eΔτH^|xN1x2p1p1|eΔτH^|x1x1x0

Note here we have the periodic boundary condition(PBC): $x{N+1}=x{0}=x$. This comes from the Boson statistics. And for Fermions we will later see a anti-PBC.

Now we have to consider elements of the type :

xk+1pkpk|eΔτH^(x^,p^)|xk=12πeipkxk+1pk|eΔτH(xk,pk)+O(Δτ2)|xk=12πexp{Δτ[pk22mipkxk+1xkΔτ+V(xk)+O(Δτ)]}

Here is defined previously.

So the partition function is:

Z=limN[i=1Ndxidpi2π]exp{1k=1NΔτ[pk22mipkxk+1xkΔτ+V(xk)]}|xN+1=x1

The integral over the momenta is Gaussian, and can be calculated directly:

dpk2πexp{Δτ[pk22mipkxk+1xkΔτ]}=m2πΔτexp[m(xk+1xk)22Δτ]

Then the partition func becomes:

Z=limNi=1Ndxi2πΔτ/m]exp{1j=1NΔτ[m2(xk+1xkΔτ)2+V(xk)]}|xN+1=x1

Note that here we have a factor which is divergent and independent of :

C(m2πΔτ)N/2=exp[N2ln(mN2π2β)]

Whene is big, this factor becomes exponetially big. However we do not need to warry because this could be cancelled out. Finally the is written as a functional integal of :

(30)Z=Cx(β)=x(0)Dxexp{10βdτ[m2(dxdτ)2+V(x)]}

From quantum mechanical path integral , to finite temperature integral:

  • (i) Wick rotation: .
  • (ii) $\mathcal{L}{E}=-\mathcal{L}{M}(\tau \equiv i t)=\frac{m}{2}\left(\frac{d x}{d \tau}\right)^{2}+V(x)$.
  • (iii) .
  • (iv) Require periodicity: .

With these steps the int becomes (30), which is called imaginary-time formalism:

exp(idtLM)exp(SE)exp(10βdτLE)

E: Euclidean.
M: Minkowski.

For fields with non-zero spin: few modification, still works.


To compute (30), write into Fourier sum:

(33)x(τ)=1βn=xneiωnτ

with periodic conditions, $\omegan = 2\pi n /\beta \hbarx(\tau)x_n^* = x{-n}xxn = a_n +ib_n, a_n = a{-n}, bn = -b{-n}$,

(34)x(τ)=1β{a0+n=1[(an+ibn)eiωnτ+(anibn)eiωnτ]}

Here, is called the amplitude of Matsubara zero mode.

Generally with the Fourier sum , the quadratic structure int can be calculated:

10βdτx(τ)y(τ)=1β2m,n=xnym10βdτei(ωn+ωm)τ=1β2m,n=xnymβδn,m=1βn=xnyn

Use this the int in is:

10βdτm2[dx(τ)dτdx(τ)dτ+ω2x(τ)x(τ)]=m2βn=[iωniωn+ω2]xnxn=m2βn=(ωn2+ω2)(an2+bn2)=m2βω2a02mβn=1(ωn2+ω2)(an2+bn2)

Next we need to consider the part. With , change the variables to and ,

Dx(τ)=|det[δx(τ)δxn]|da0[n=1dandbn]

The determinant is hard to compute and could be written into the previous hard-to-compute :

CC|det[δx(τ)δxn]|

Now we have the partition function:

Z=Cda0[n=1dandbn]exp[m2βω2a02mβn=1(ωn2+ω2)(an2+bn2)]

This is a Gaussian integral, and finally:

(42)Z=C2πβmω2n=1πβm(ωn2+ω2),ωn=2πnβ

Calculate :