# Essential background for QFT study

The preface to Mark Srednicki's "Quantum Field Theory" says that to be prepared for the book, one must recognize and understand the following equations:

$$\frac{d\sigma}{d\Omega} = |f(\theta,\phi)|^2, \qquad (1)$$ $$a^{\dagger}|n\rangle = \sqrt{n+1} \space |n+1\rangle, \qquad (2)$$ $$J_{\pm} |j,m \rangle = \sqrt{j(j+1)-m(m\pm 1)} \mid j,m \pm 1 \rangle, \qquad (3)$$ $$A(t) = e^{+iHt/\hbar}Ae^{-iHt/\hbar}, \qquad (4)$$ $$H = p\dot{q}-L, \qquad (5)$$ $$ct'=\gamma (ct-\beta x), \qquad (6)$$ $$E=(\mathbf{p}^2c^2+m^2c^4)^{1/2}, \qquad (7)$$ $$\mathbf{E} =-\mathbf{\dot{A}}/c-\mathbf{\nabla} \varphi. \qquad (8)$$

I am certainly not ready to dive into this book, so I would like some help identifying these equations and learning more about their fundamental usefulness.

I don't recognize (1), but (2) looks like a quantum mechanical creation operator? I thought those were only really useful in the context of the harmonic oscillator problem, but maybe everything is just a complicated HO problem? (3) has to do with angular momentum? (4) is a plane wave solution to the Schrodinger Eqn? (5) is the classical mechanics Hamiltonian, with cannonical coordinates? (6) is the relativistic Lorentz transformation. (7) is the general form of mass energy equivalence from SR. (8) is the electric field expressed as vector and scalar potentials? Is that really the only E&M machinery required?

Any insight as to why these particular expressions are relevant / important / useful to QFT is also appreciated. Also, where are the statistical mechanics ideas hiding? In the QM?

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My impression is that authors just threw out a random list of basic topics that should give you a rough idea of the level required. They are definitely not everything you need and likewise, not all of those points are required equally; e.g. one can do a good portion of QFT without ever talking about EM -- one would miss a lot though (including all the motivation). –  Marek Jul 20 '11 at 20:28
In other words, he wants you to know basic QM, SR and EM (first formula comes from the scattering theory that you might or might not have encountered in advanced QM). These are the standard prerequisities in most of the books. If you feel shaky in any of those subjects, it might help to review a bit. –  Marek Jul 20 '11 at 20:36
Related question on Math.SE: math.stackexchange.com/q/1086169/11127 –  Qmechanic Dec 31 '14 at 0:37

They are:

1. The definition of the scattering cross-section in terms of the scattering amplitude.

2. HO/photon creation operator.

3. Angular momentum raising and lowering operator.

4. Heisenberg equations of motion for an operator.

5. Definition of the Hamiltonian.

6. Lorentz transformation.

7. Relativistic equation for energy.

8. Electric field in terms of scalar and vector potentials.

(1) is because you calculate a lot of cross-sections, (2) & (3) because you use a lot of raising and lowering operators, (4) because you need to know the difference between the Schrodinger, Hisenberg, and Interaction pictures of QM, (5) becuase you use both Lagrangians and Hamiltonians, (6) & (7) for basic special relativity, and (8) for basic E&M.

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Great answer, thanks! –  Greg L Jul 21 '11 at 14:11

''where are the statistical mechanics ideas hiding? In the QM?''

Mark Srednicki's "Quantum Field Theory" is, strictly speaking, only about relativistic quantum field theory at zero absolute temperature.

Thus the book doesn't contain any statistical mechanics. Therefore a corresponding formula, perhaps $\rho=e^{-S/k_B}$, is missing from the prerequisites.

However, quantum fields are heavily used in statistical mechanics, as calculations in many-body problems are far more easily represented in terms of quantum fields than in terms of multiparticle states. These quantum fields are often nonrelativistic (which simplifies things a lot, as all renormalizations are finite), but for applications to heavy ion collisions and cosmology, one also needs relativistic statistical mechanics.

Of course, the techniques of quantum field theory as presented in Srednicki and of quantum field theory as presented in a statistical physics book (Reichl or Umezawa, say) are related, and indeed, each side borrows heavily from the other.

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