# Vacuum to Vacuum Amplitudes and Functional Integrals

1. Oct 13, 2005

### robousy

Hi,
I am reading chapter 5 of Ryder regarding path integrals and vacuum to vacuum transition amplitudes in presence of source.
I follow the math but don't have a clear physical picture.
The formula is:
$$Z[J]=\int Dq \: exp ( \frac{i}{h}\int dt(L+hJq+\frac{1}{2}i\epsilon q^2) )$$
Can someone explain what this is the transition amplitude of please?
I think its saying:
1) pick a point in space
2) overlay a source (eg EM field)
3) A particle may be raised above the vacuum ground state at some point but ultimately at the beginning and end of time the vacuum will stay the vacuum - ie the vacuum will never turn into a stable particle.
I don't really think this is correct so please correct me!
:)

2. Oct 14, 2005

### hellfire

IMHO, this formula is of the generating functional, which is not the transition amplitude. However, the n-th functional derivative with respect to J of the generating functional gives you the n-point Green function. The n-point Green function is related to the transition amplitude of a scattering event involving n particles, i.e. it is related to one element of the S-matrix <in|out>, which can be written as a function of creation and annihilation operators acting on <0| and |0> (which in turn can be expressed as a time ordered product of fields).

3. Oct 14, 2005

### dextercioby

Two words for ya: Read Zee !!

Daniel.

4. Oct 14, 2005

### Hans de Vries

This is basically the start formula of chapter 6 which denotes the
path-integral for scalar fields.

L is Lagrangian which is proportional to the amount of phase-changes
over the trajectory of the particle as a result of it's rest mass and motion.

The J term takes account of the phase changes over the trajectory
of the particle as a result of the Electric and Magnetic Aharonov Bohm
effects. (The EM interactions)

Regards, Hans

Last edited: Oct 14, 2005
5. Oct 14, 2005

### robousy

Thanks Hans.

What do you mean phase change over trajectory - the phase change of what.

I can conceptually picture phase changes for things like EM waves for example but I have a block when we talk about the phase of a particle.

6. Oct 14, 2005

### robousy

Thanks.

The QFT books I currently learn from are:

Ryder
Mandl and Shaw
Peskin and Schr.
Weinberg - Quantum theory of fields.

Do you feel that Zee is sufficiently superior to these that it is worth purchasing in addition?

If so I will certainly purchase it.

7. Oct 15, 2005

### Hans de Vries

If you go to Peskin & Schroeder chapter 9 then you'll find an introduction in
terms of the "sum over phase changes". A very popular introduction on the
elementaries of this is Feynman's "QED The strange theory of matter and light"

Be aware that not all math is what it seems in these texts, e.g: in $\langle x_b | e^{-iHt/\hbar} | x_a \rangle$
the Hamiltonian H is an operator (it includes differentiation) which makes the
whole exponent an operator.

Regards, Hans.

8. Oct 16, 2005

### robousy

Ok, thanks Hans. I'll check Peskin.

9. Oct 17, 2005

### dextercioby

It's difficult to classify books & say one is superior to another, but i'll tell you that Zee's book explains the physics behind the path integral and what those "source terms" in the generating functionals mean and their connection with vacuum fluctuations.

Daniel.

10. Oct 17, 2005

### robousy

thx dextercioby. A friend has Zee so I'll just borrow.

Incidently. Its pretty obvious that you have a really good grasp of anything quantum. (qft qm, all the maths) and probably a bunch of other stuff.