Register to reply

Expansion around a classical vacuum

by GargleBlast42
Tags: classical, expansion, vacuum
Share this thread:
GargleBlast42
#1
May15-11, 05:26 PM
P: 28
Hi everyone,

I have a severe confusion about the notions of "expanding the theory around a classical vacuum" and "considering small fluctuations around a classical vacuum" which I find in QFT textbooks.

My problem is: in the path integral [tex]\int D\phi e^{i S[\phi]}[/tex] one doesn't integrate only over field configurations close to the vacuum, but over all field configurations. And when one is considering a perturbative expansion, this expansion is in the coupling constant (like [tex]\lambda[/tex] in [tex]\phi^4 [/tex] theory), but one doesn't assume [tex]\phi[/tex] to be small, or am I wrong?

So the questions would be: Why does one require the field configurations to be small fluctuations around a classical vacuum? And what would happen if I was expanding the theory about a field configuration that is not a classical vacuum (except that the mass could be possibly negative)? The first question is more important for me.

I would be very grateful for any clarification.
Phys.Org News Partner Physics news on Phys.org
First in-situ images of void collapse in explosives
The first supercomputer simulations of 'spin?orbit' forces between neutrons and protons in an atomic nucleus
Magnets for fusion energy: A revolutionary manufacturing method developed
GargleBlast42
#2
May17-11, 02:45 PM
P: 28
I'm sorry for bumping this, but I would be really happy about any input.
Demystifier
#3
May18-11, 03:23 AM
Sci Advisor
Demystifier's Avatar
P: 4,569
You are right that one integrates over all values of fields, not only the small ones. The assertion that field is small means something else. It refers to a physical value of field, such as the boundary value appearing in the definition of the path integral. In particular, if you calculate the vacuum-to-vacuum transition, then the boundary values of the field are zero, which, of course, are small.

genneth
#4
May18-11, 05:14 AM
P: 981
Expansion around a classical vacuum

Do you have any idea how to actually compute these integrals? If not, I'm afraid that the answer won't make sense --- the entire apparatus is rather formal, which is to say, it is a series of methods to circumvent the problem that evaluating these integrals exactly is impossible.
Avodyne
#5
May18-11, 04:20 PM
Sci Advisor
P: 1,190
There is a strong analogy with evaluating an ordinary integral of this type by the method of stationary phase. One first finds the point(s) of stationary phase, and then approximates the integral as a gaussian (which equates to treating the fluctuations as "small" in some formal sense) around each such point. Corrections to the gaussian correspond to doing perturbation theory in QFT.


Register to reply

Related Discussions
Expansion of gas into a vacuum Classical Physics 0
Classical Version of Vacuum Polarization Quantum Physics 5
Expansion into a vacuum General Physics 2
The classical aether vs. the modern vacuum Special & General Relativity 57
Expansion of a Gas into a Vacuum Engineering, Comp Sci, & Technology Homework 1