# $\langle B|A \rangle$ expressed in terms of the Partition Function

Say you have an electron departing from point A and reaching poing B after a time t.

According to some helping friend, the Partition Function for that electron going from point A to B can be written as

$$Z = \int_{A \to B} [\mathcal{D}x]~ e^{iS[x]}$$

where $\mathcal{D}x$ is the measure that sums up over all paths going from $A$ to $B$, and $e^{iS[x]}$ is the weight of each path, $S[x]$ is the action.

That friend states then "From this partition function all desirable quantities can be obtained."

Not having much idea about Feynman's Path Integral Formulation of Quantum Mechanics, I have looked around a bit, and I would like someone to confirm the following statement I make:

The amplitude for the electron to go from A to B in the time t can be found in terms of that Partition Function $Z$ given above, as

$$\langle B|e^{-iHt}|A\rangle = Z$$

Did I catch it right?

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Of course, if there are several answers, I will choose whichever has more additional information, but in principle, a simple confirmation (better with some reference or link to further reading) is enough. –  Mephisto May 11 at 1:21
$Z$ is related to e.g. $<A|B>$ by the LSZ reduction formula. Maybe someone can explain this formula... –  innisfree May 11 at 1:30

For simplicity, let's restrict the discussion to that of a single particle moving in one dimension. Path integrals can be performed in much broader contexts like quantum field theory, but I think that would conceptually obscure the issue at this point.

Let $H$ denote the (time-independent) quantum hamiltonian. Then the time evolution on the system is governed by the operator $U(t) = e^{-itH/\hbar}$ called the time evolution operator. If at a time $t_a$ the particle is in the state $|x_a\rangle$, the eigenstate of the position operator corresponding to position $x_a$, then the probability amplitude for finding the particle in the state $|x_b\rangle$ at some later time $t_b$ is given by $$K(t_b,x_b;t_a,x_a) = \langle x_b|U(t_b-t_a)|x_a\rangle$$ This object is often referred to as the propagator. It turns out the propagator has the following path integral expression $$K(t_b,x_b;t_a,x_a) = \int \mathcal Dx\mathcal Dp\, e^{\frac{i}{\hbar}S}$$ where the integral is over all paths $(x(t), p(t))$ in classical phase space satisfying the boundary conditions $x(t_a) = x_a$ and $x(t_b) = x_b$.

Now you could ask, what about if at time $t_a$, the system started in a state $|\psi_a\rangle$ that is not necessarily a position eigenstate, and I want to know what the probability is of finding that particle in the state $|\psi_b\rangle$ at time $t_b$. Can I write this in terms of the path integral? Yes!

What we're looking to compute is the following quantity: $$\langle \psi_b|U(t_b-t_a)|\psi_a\rangle$$ Now we recall that the we can write the identity operator as $$I = \int dx\,|x\rangle\langle x|$$ Inserting this expression twice into the amplitude we want to compute, and using integration variables $x_a$ and $x_b$ gives \begin{align} \langle \psi_b|U(t_a-t_a)|\psi_a\rangle &= \int dx_a\int dx_b\langle \psi_b|x_b\rangle\langle x_b|U(t_a-t_a)|x_a\rangle\langle x_a|\psi_a\rangle \\ &= \int dx_a\int dx_b\langle \psi_b|x_b\rangle\langle x_a|\psi_a\rangle K(t_b,x_b;t_a,x_a) \end{align} Now we just write the propagator in terms of the path integral as above to obtain $$\langle \psi_b|U(t_a-t_a)|\psi_a\rangle =\int dx_a\int dx_b\langle \psi_b|x_b\rangle\langle x_a|\psi_a\rangle \int \mathcal Dx\mathcal Dp\, e^{\frac{i}{\hbar}S}$$

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Thus, that so-called partition function equals the path integral expression of the propagator (at least in the restricted 1-dimensional single-particle case), and the statement in my question is right when $|A>$ and $|B>$ are eigenstates of position, right? –  Mephisto May 11 at 2:16
@Mephisto Basically yes. –  joshphysics May 11 at 2:23
Thanks! I have to see now what is that $\int \mathcal Dx \mathcal Dp$ but I can cope with that, I see in Zee/Srednicki that it is a short notation for an integral with multiple products due to dividing the path in small time steps... I'll have a closer look. Thanks! –  Mephisto May 11 at 2:36
What your friend actually meant is that you can obtain all desirable correlation functions. Assuming you're talking about a non-relativistic electron, consider a source term added to your action $$S'[x] = S[x] + \int dt \, J(t) x(t)\,.$$
Now you can write any correlation function as a derivative of $\ln Z$ calculated at $J = 0$, i.e. $$\langle B| \hat{x}(t') | A \rangle = \left.\frac{\partial}{\partial J(t')} \ln Z\right|_{J=0}\,,$$ $$\langle B|\hat{T}\left\{ \hat{x}(t') \hat{x}(t'') \right\}| A \rangle = \left.\frac{\partial^2}{\partial J(t')\partial J(t'')} \ln Z\right|_{J=0}\,,$$ and so on, where $\hat{T}$ is the time ordering operator.
I don't know which bird dropped that shit here with no comments. It is specially annoying in this case, since any comment would help me understanding this answer. $lnZ$ is like the generator of a Lie group or something, right? I still lack the appropiate knowledge. –  Mephisto May 11 at 15:40