# Functional time-dependent perturbation theory

1. Sep 26, 2015

### ShayanJ

Today, in my advanced particle physics class, the professor reminded the time-dependent perturbation theory in NRQM and derived the formula:

$\displaystyle \frac{da_m(t)}{dt}=-i \sum_n e^{-i(E_n-E_m)} \int_{\mathbb R^3}d^3 x \phi^*_m (\vec x) V(\vec x,t) \phi_n(\vec x)$.

Then he said that this is true also for Klein-Gordon and Dirac equations which confused me. I asked him how can that be true because we assumed an equation of the form $(H_0+V)\psi = i \frac{\partial \psi}{\partial t}$ and started from eigenfunctions of $H_0$ to do the perturbation but Dirac and Klein-Gordon equations are not of this form. His answer wasn't convincing too. So I investigated a little bit and then suddenly remembered the functional Schrodinger equation. Now I think that using the functional Schrodinger equation, we can derive a similar formula:

$\displaystyle \frac{da_m(t)}{dt}=-i \sum_n a_n(t) \int D \phi \ \psi^*_m [\phi] V[\phi] \psi_n[\phi] e^{-i(E_n-E_m)}$

and interpret the $a_n$s as transition amplitudes between different field configurations. Is this true?
Thanks

2. Sep 26, 2015

### HomogenousCow

That looks like the usual Dyson series in the interaction picture, it still holds in QFT. Although you wouldn't be doing it in position space I assume.

3. Sep 26, 2015

### ShayanJ

All that I can find in books are space-time or momentum integrals. Can you give a reference that contains such a functional integral on field configurations for the dyson series?

4. Sep 27, 2015

### vanhees71

You find both, the operator and the path-integral formalism in my lecture notes on QFT:

http://fias.uni-frankfurt.de/~hees/publ/lect.pdf

and in any modern textbook on the subject. Don't read old textbooks like Bjorken/Drell vol. I. It's a very nice historical introduction to how difficult life was before the machinery of QFT was discovered, but that's it.

The functional formalism is not so often discussed in textbooks. An excellent exception is

B. Hatfield, Quantum field theory of point particles and strings, Perseus Books (1992)

5. Sep 27, 2015

### ShayanJ

So the functional formalism is old and hard and the modern ways to do things are operator and path integral formalism and the formula that I wrote is correct in...which formalism? Path integral or functional? I'm confused because the D in the integral reminds me of path integrals.

Also I've read Hatfield but there was nothing like the formula I wrote in my first post. Is there such a formula in your lectures?

Also I didn't read Bjorken and Drell. Our main text is Halzen and Martin. Its just a advanced particle physics class, not full QFT.

6. Sep 27, 2015

### vanhees71

I don't like wave functions in relativistic QT. The formula is, of course, somewhere in operator form in my lectures (it's the usual Dyson-Wick series, truncated at the first order). I'm not used to the functional formalism. I don't know, whether it's in much use by specialist theorists in QFT. In our field of research (relativistic heavy-ion collisions) we use either the operator or the path-integral formalism and, at the end, the Feynman rules of perturbation theory derived from there. I like Hatfields book, because he carefully explains, how to calculate things in all the three formalisms. I guess, you should find a derivation of the (perturbative) S-matrix elements in all three formalisms in his book.