Path integral and partition function

1. Jan 25, 2012

kof9595995

I have some confusions identifying the following objects:
(1)Some transition amplitude involving time evolution(Peskin page 281, eqn 9.14):
$$\langle\phi_b(\mathbf x)|e^{-iHT}|\phi_a(\mathbf x)\rangle=\int{\cal D\phi \;exp[i\int d^4x\cal L]}$$
(2)Partition function(after wick rotation)
$$Z_0=Tr(e^{-\beta H})=\int{\cal D\phi \;exp[i\int d^4x\cal L]}$$
(3)Functional determinant(Klein-Gordon for example, Peskin page 287, eqn 9.25)
$$const\times [det(\partial^2+m^2)]^{-\frac{1}{2}}=\int{\cal D\phi \;exp[i\int d^4x\cal L]}$$
All three appear in chap 9 of Peskin's textbook. though (2) is not explicitly written.
I can convince myself (2) and (3) are the same, but have trouble with (1). To make LHS of (1) the same with LHS of (2), shouldn't we impose periodic boundary condition on (1) and integrate over all initial states? That is,
$$\int{\cal D}\phi_a\langle\phi_a(\mathbf x)|e^{-iHT}|\phi_a(\mathbf x)\rangle=\int{\cal D}\phi_a\int{\cal D\phi \;exp[i\int d^4x\cal L]}$$
But then the RHS of (1) and (2) become different.

Last edited: Jan 25, 2012