Interpretation of the functional Z (in Zee).

In summary, Zee finds that he can write the amplitude for a ground state wave function given by $$\langle q_f|e^{-iHT} |q_i\rangle = \int Dq(t) e^{iS} $$He then observes that by inserting a complete set of states we can write$$Z = \int q_f \int q_i \langle 0 | q_f\rangle \langle q_f | e^{-iHT} |q_i\rangle \langle q_i |0 \rangle = int q_f \int q_i \psi_0^* (
  • #1
center o bass
560
2
In Zee's book at page 12 in both editions he finds that he can write the amplitude

$$\langle q_f|e^{-iHT} |q_i\rangle = \int Dq(t) e^{iS} $$

where T is the time between emission at ##q_i## and observation at ##q_f##. He then states that we often define

$$Z = \langle 0 | e^{-iHT} |0 \rangle $$

And he observes that by inserting a complete set of states we can write

$$ Z = \int q_f \int q_i \langle 0 | q_f\rangle \langle q_f | e^{-iHT} |q_i\rangle \langle q_i |0 \rangle = int q_f \int q_i \psi_0^* (q_f) \psi_0 (q_i) \langle q_f | e^{-iHT} |q_i\rangle. $$

But then a few sentences down he writes

$$Z = \int Dq(t) e^{iS}. $$

Is there an error in Zee here or ##Z## as defined above actually equal to the path integral given by ##\langle q_f|e^{-iHT} |q_i\rangle##? If so, how does the integration over the ground state wavefunction vanish?
 
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  • #2
If so, how does the integration over the ground state wavefunction vanish?
I don't see how are you coming to this conclusion.
 
  • #3
Zee often leaves out details like this. He is presumably now defining Dq to include the integration over the ground-state wave function. For a more thorough explanation of how this works, see Srednicki's text.
 
  • #4
If I remember right, Zee leaves out the crucial detail to introduce a little imaginary part to the Hamiltonian such that you get the vacuum-to-vacuum-transition amplitude, i.e., in the path integral contributions from all other states are exponentially damped out in the limit [itex]t_i \rightarrow -\infty[/itex] and [itex]t_f \rightarrow \infty[/itex], where [itex]t_i[/itex] and [itex]t_f[/itex] are the initial and final time in the original path integral for the time-evolution kernel.

A very good treatment of this issue can be found, e.g., in

Bailin, Love, Gauge Theories
 
  • #5
andrien said:
I don't see how are you coming to this conclusion.

I did not mean "vanish" in the sense of becoming zero, but in the sense of disappearing from the expression :)
 
  • #6
vanhees71 said:
If I remember right, Zee leaves out the crucial detail to introduce a little imaginary part to the Hamiltonian such that you get the vacuum-to-vacuum-transition amplitude,
I thought a substitution t→it is done so as to give a proper meaning to the exponential
exp(iS),not to hamiltonian.
 

1. What is the functional Z in Zee?

The functional Z in Zee refers to a mathematical concept used in quantum field theory. It is a functional integral that is used to calculate the probability amplitude of a quantum field theory.

2. Why is the functional Z important in quantum field theory?

The functional Z is important in quantum field theory because it allows us to calculate the probability amplitudes of various processes and interactions in a quantum field. This information is crucial in understanding the behavior of particles and the laws of nature at a fundamental level.

3. How is the functional Z calculated?

The functional Z is calculated using functional integration, which is a mathematical tool used to integrate functions of functions. In quantum field theory, this involves summing over all possible configurations of the quantum field to calculate the probability amplitude.

4. What is the significance of the functional Z in particle physics?

The functional Z in particle physics allows us to calculate the probability amplitudes for various particle interactions and processes. This is important in understanding the behavior of particles and the fundamental forces that govern their interactions.

5. Can the functional Z be applied to other fields of science?

While the functional Z is primarily used in quantum field theory, it has also been applied to other areas such as statistical mechanics and economics. It is a powerful mathematical tool that can be used to analyze complex systems and calculate probabilities in various fields of science.

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