## Vacuum to Vacuum Amplitudes and Functional Integrals

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!
:)
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 Blog Entries: 9 Recognitions: Science Advisor 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 , 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).
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## Vacuum to Vacuum Amplitudes and Functional Integrals

 Quote by 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: $$Z[J]=\int Dq \: exp ( \frac{i}{h}\int dt(L+hJq+\frac{1}{2}i\epsilon q^2) )$$

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

 Quote by Hans de Vries 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.
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.

 Quote by dextercioby Two words for ya: Read Zee !! Daniel.

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.

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 Quote by 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.
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.
 Ok, thanks Hans. I'll check Peskin.

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