Given a generic Quantum Field Theory, I am able to compute only two physical quantities: the decay rate of particles and cross sections of interactions. Does exist other physical observables which can be computed and are physical relevant?
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$\begingroup$ There are plenty of relevant observables that can be computed, and many of them can be measured, either directly or indirectly. There is not much difference between quantum mechanics and qft with respect to this aspect. $\endgroup$– yuggibJan 13, 2017 at 12:23
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1$\begingroup$ Just to give a concrete example, which you can find computed in textbooks: the anomalous magnetic moment of the electron. $\endgroup$– pppqqqJan 13, 2017 at 12:26
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2$\begingroup$ Possible duplicate of Is the S-Matrix the only quantum field observable? $\endgroup$– AccidentalFourierTransformJan 13, 2017 at 12:40
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$\begingroup$ You are right: I forgot the anomalous magnetic moment that you can compute with loops contributions, and there is also Casimir effect; anyway I can't find other relevant physical quantities. $\endgroup$– YildizJan 13, 2017 at 12:40
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$\begingroup$ I wouldn't say it's a duplicate; we are using the same word "observable" with two different meaning $\endgroup$– YildizJan 13, 2017 at 12:42
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1 Answer
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By using QFT we calculate the Green functions and vertices, perturbatively or non-perturbatively. The Green function is the quantity $$ G(x_{1},...,x_{n}) = \langle \text{vac}| T(\prod_{i}\varphi_{i}(x_{i})|\text{vac}\rangle, $$ where $\varphi_{i}$ is the given field in Heisenberg representation and $T$ is chronological ordering, while the vertex is $$ \Gamma^{(n)}(x_{1},...,x_{n}) = \frac{\delta^{n}\Gamma[\varphi]}{\delta \varphi_{1}(x_{1})...\delta\varphi_{n}(x_{n})}\bigg|_{\varphi = 0}, $$ where $\Gamma[\varphi]$ is the quantum effective action functional.
They give us the information about:
1) Probabilities of transitions (cross-sections, decay rates). For example, one may compute the vertex $\Gamma_{\pi^{0} \gamma\gamma}$ where $\pi^{0}$ is the pion and $\gamma$ is the photon in zero-mass limit $m_{\pi} = 0$, and find that it is non-zero when loop corrections are given into account;
2) Poles and their position (i.e., the set of particles with which are created by the given fields), i.e., particles spectrum and their masses. This is reflected in spectral representation of propagators. Typical example is the Goldstone theorem, which states that in the theory with spontaneously broken symmetry $G$. Assume the commutator $\langle \text{vac}|[J^{\mu}(x),\varphi_{n}(y)]|\text{vac}\rangle$, where $J^{\mu}$ is conserved current of the symmetry $G$ and $\varphi_{n}$ is the set of fields with given non-zero charge. Then the Goldstone theorem says that $$ \langle \text{vac}|[J^{\mu}(\mathbf{x},t),\varphi_{n}(\mathbf{y},t)]|\text{vac}\rangle = i\delta(\mathbf x- \mathbf y)\int d\mu^{2}\rho(\mu^{2}), $$ where the spectral function $\rho(\mu^{2})$, containing the information about poles, is non-zero for zero $\mu^{2}$ as long as $G$ is spontaneously broken. This means that there are massless particles.
3) Running couplings and couplings correction (muon anomalous magnetic moment, QCD running coupling);
4) Symmetries (whether they are hold in the quantum field theory). For example, if we find $\langle \sigma \rangle \neq 0$ in the theory with classical action having $Z_{2}$ symmetry $\sigma \to -\sigma$ means that the symmetry is broken spontaneously. If we find the operator quantity $$ \partial_{\mu}\langle J^{\mu}\rangle =\text{A} \neq 0 $$ in a theory with classically conserved current $J^{\mu}$, then the corresponding symmetry is explicitly broken.
