Surface terms for field path integrals?

Question

My question relates to something that I´ve seen in many books and appears in all its glory here: Ryder, pg 198

My question is about eq. 6.74. Which I repeat below:

$$i \int {\cal D}\phi \frac{\delta \hat{Z} [\phi] }{\delta \phi} exp \left(i\int J(x) \phi(x) dx\right) = i\; exp \left(i\int J(x) \phi(x) dx\right) \hat{Z}[\phi] \Bigg|_{\phi\rightarrow\infty}+ \int {\cal D}\phi J(x) \hat{Z}[\phi] exp \left(i\int J(x) \phi(x) dx\right)$$

$\phi$ is a scalar field, J is a source, $x = x_{\mu}$ in 4D Minkowsky space and $\hat{Z}[\phi] = \frac{e^{iS}}{\int {\cal D}\phi\; e^{iS}}$

The author is clearly doing a integral by parts and the first term on the right hand side is a kind of surface term for the path integral. He then considers this term to be zero and the second one gives us:

$$i \int {\cal D}\phi \frac{\delta \hat{Z} [\phi] }{\delta \phi} exp \left(i\int J(x) \phi(x) dx\right) = J(x) Z[J]$$

The trick thing here is that integral limits for $\int{\cal D}\phi$ are not very obvious (at least not to me). You are in fact summing up for all field configurations. So, there are actually two problems in my mind:

1. For what configuration of $\phi$ is the surface term calculated? (the author says it is $\phi \rightarrow \infty$)

2. Assuming the author is right about taking huge $\phi$: why is this term zero?

This applies to path integrals in general: can we do the usual trick of throwing out surface terms safely?

Answer 1

One mustn't confuse field space with physical space. The field space is some sort of manifold without boundaries (for a nonlinear sigma model), or $R^n$ for usual field theories, in either case, integration by parts works in Euclidean space, or if you add a little imaginary part to the propagators so that the action is decaying at large values of $\phi$.

The integration by parts in field space is simple--- there are no boundaries in field space, except at infinite field values, and the Euclidean or slightly off-Minkowsky action decays at infinity.

There is no relation to the integration by parts in physical space involved for instantons or other topological things.

Answer 2

He is imagining the possible values of $\phi$ as a Euclidean space $\mathbb{R}^n$ with a "boundary sphere" $S^{n-1}$ at infinity. This is a reasonable thing to think about since we usually consider a UV cut-off and consider the limit as this cut-off goes to infinity. (This is just the ordinary way of thinking about integrals on noncompact spaces.)

This boundary term is zero when we assume that the action decays quickly enough as $\phi$ grows large. This assumption is necessary for his argument to work.

I hope this helps. I have always been a bit weary about these integration by parts arguments myself. I guess physics texts like these just always assume Stokes' law works.

These sorts of arguments can be subtle when the target of $\phi$ is not a contractible space like $\mathbb{R}^n$. For example, if $\phi$ is a map from $\mathbb{R}^4$ to $SU(2)$, which is topologically a 3-sphere, there are topologically nontrivial boundary configurations for $\phi$ labelled by $\pi_3(SU(2))=\mathbb{Z}$ . This is the case when one has Yang-Mills instantons.

One also must be careful of course when the spacetime has boundary. For example, when computing a propagator, we have a spacetime that looks like a tube: $M \times \mathbb{R}$, and we use the boundary condition that on one end of the tube we have one state and on the other end we have another state. Then the path integral computes the propagator between the states.