Path integral and partition function

[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.

 

[azntoon]

No, the right-hand sides of (1) and (2) are still the same. The left-hand side of (1) is just a special case of the left-hand side of (2), where the initial state is a single wavefunction, rather than a linear combination of many wavefunctions. In this case, the integration over all possible initial states can be simplified to a single integration over the initial wavefunction.