# What does the concept of phase space mean in particle physics?

I came across the concept of phase space in statistical mechanics.

How does this concept come about in particle physics?

Why was it introduced and how is it used?

What does it mean when textbooks say there is no enough phase space for that reaction to take place?

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Hah! As used in particle physics it generally means a scalar representing the volume of phase space available to some process and not the space itself as is often used in classical mechanics. If I find the time and motivation I may write an actual answer later. – dmckee Aug 7 '12 at 0:33
I've written two earlier answers that touch on this, without quite covering all the bases here: physics.stackexchange.com/a/11671/520 and physics.stackexchange.com/a/31517/520 . – dmckee Aug 7 '12 at 0:39

All of particle physics pretty much comes down to calculating decay rates or scattering cross sections. In either case, the quantity you calculate is a function of the momenta of the initial and final particles.

For the initial particle(s), you know the momenta, because they're determined by the design of your particle accelerator or whatever experiment you've set up. But for the final particles, there are usually a variety of different momenta you can have. The phase space is just the mathematical space of all possible momenta of all the outgoing particles.

As a very simple example, consider $\mathrm{e}^-\mathrm{e}^+\to\gamma\gamma$. At LEP, each of the electron and positron would have a momentum of $104.5\text{ GeV}/c$, which means each of the photons would have essentially that same magnitude of momentum. But the photons could come out in any direction, subject only to the restriction that they must emerge back-to-back. So in this example, you have a two-dimensional space of possible momenta (namely $104.5\text{ GeV}/c$ in any direction) that the photons could have. That space is what particle physicists call the phase space.

The reason this term comes up is that when you calculate something in quantum field theory, it involves doing an integral over phase space. This might be written as something like

$$\int\mathrm{d}^3\vec{p}_3\mathrm{d}^3\vec{p}_4\cdots\delta(\vec{p}_1 + \vec{p}_2 - \vec{p}_3 - \vec{p}_4 - \cdots)[\text{stuff}]$$

Basically you integrate over all possible values of the momenta of the outgoing particles, subject to conservation of momentum (that's the delta function) and any other constraints that may come out of the calculation. (There are also weighting factors that I've omitted.)

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Nice. Saves me the trouble of getting my thoughts into proper order to write something coherent. – dmckee Aug 7 '12 at 14:46