Abstract
This chapter starts with Wigner’s classification of particles based on the unitary representations of the Poincaré group and the classification of field variants as enforced by Lorentz symmetry. Generic requirements of action functionals in a (quantum) field theory are formulated. Next appropriate actions are derived for the various field variants. By building Lorentz scalars from the fields and their derivatives it is shown that the free-field part for a spin-0 and spin-1/2 field is fixed from Poincaré symmetry and arguments of dimensional renormalizability. The quantization of these scalar and spinor field theories is sketched. The central theme in fundamental physics is the idea of a gauge theory – or – Yang-Mills theory. Gauge fields are introduced in order to render global phase symmetries of wave functions spacetime-dependent. Both the coupling of these spin-1 fields to the spinors and their kinetic energy term in a Lagrangian are largely “dictated” by the local internal symmetry. It is shown how the Klein-Noether identities fix the action for any Yang-Mills theory (with mild conditions on the symmetry group). Field equations for higher-spin fields are mentioned. A further subsection deals with spontaneous symmetry0 breaking which arises in a (quantum) field theory if a symmetry of an action is no longer a symmetry of the “ground state”. The mechanism by which in a Yang-Mills gauge theory would-be Goldstone bosons disappear at the expense of massive gauge bosons is explained. In a further chapter the discrete symmetries related to space inversion, time reversal, and charge conjugation, as well as the CPT theorem are dealt with. The last section refers to effective field theories, a notion which is becoming more and more accepted. Among other things, it allows to relate theories which apply to phenomena on different levels of granularity or on different length scales. An important technical means is the approach of the renormalization group flow and the notion of \(\beta \) functions.
I was, at one time, greatly interested in establishing all linear equations which are invariant under the inhomogeneous Lorentz group \(\ldots \)
E. P. Wigner in “Invariant Quantum Mechanical Equations of Motion”, in: Theoretical Physics Lectures presented in Trieste, Italy; International Atomic Energy Agency, Vienna 1963, 59–82.
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Notes
- 1.
A short and thorough historical reflection is Chap. 1 in [536].
- 2.
You may wonder why the factors with the imaginary unit are kept in these expressions. Indeed for unitary operators \(\,U(\Lambda ,a)\,\) they cancel on both sides of these relations. Later, however, when dealing with discrete symmetries in Sect. 5.5, we find that these factors are the key for deciding about unitarity and anti-unitarity.
- 3.
Again, the sign in front of \(\Sigma \) is a convention.
- 4.
Equivalently one could choose the Casimirs \((\vec {J}^2-\vec {K}^2)\) and \(\vec {J}\cdot \vec {K}\).
- 5.
In this Chapter I completely suppress the “hats” for operators. The context should make clear of whether an object is an operator or not.
- 6.
This interpretation is due to W. Pauli and V. Weisskopf [408]. Their work not only showed that the particle/anti-particle notion is inherent to quantum field theory–and not bound to exist only for electrons and positrons–but also explained why previous unsuccessful attempts to interpret the zero component of the Noether current as a probability density had failed.
- 7.
These are two-component fields and should not be confused with the four-component spinors introduced in the next subsection.
- 8.
Conclusive results should come from experiments on double beta decay; see 6.4.2.
- 9.
This ranges from the dynamo on your bicycle to fusion processes in the sun and to the atomic bomb.
- 10.
from the theatre of Pompey at the moment of Caesar’s death to the black hole horizon of the Andromeda galaxy at the moment of last scattering, say.
- 11.
More about “constrained dynamics” in Appendix C.
- 12.
not to be confused with the ghosts in the Faddeev-Popov and BRST quantization.
- 13.
In those cases where I use the term YM-like theories, I assume a theory with a quadratic term in the gauge field strengths, as it is derived below. In this sense, also electrodynamics is a YM-like theory.
- 14.
Utiyama investigated among others the Lorentz group as a local symmetry group. More about this in Sect. 7.6.3.
- 15.
Here I cite directly [536], Sect. 15.2, adapted to my notation.
- 16.
Its quite remarkable that these no-go theorems can be circumvented in (A)dS spacetime.
- 17.
Indeed in 2008 Nambu received the Nobel prize in physics “for the discovery of the mechanism of spontaneous broken symmetry in subatomic physics”.
- 18.
All six physicists were jointly awarded the 2010 J. J. Sakurai Prize for Theoretical Particle Physics.
- 19.
I apologize for this abundance of letters S used with different fonts. It may be pedantic, but we must conceptually distinguish the symmetry transformations from their unitary representations and their realizations as e.g. matrix multiplication on the fields. The expression (5.96) is comparable to (5.21) and (5.22).
- 20.
This was proposed as early as 1941 by E. Stückelberg.
- 21.
Although, as argued in Sect. 5.6.3, QED does not really exist on its own.
- 22.
At that time, one did not know that there are three varieties of neutrinos and their antiparticles.
- 23.
There are other regularization techniques as well. The one mostly used today is dimensional regularization, where one first treats the theory in \(\,(d-\epsilon )\) dimensions and then takes the limit \(\,\epsilon \rightarrow 0\).
- 24.
The group acting here is simply the additive group of transformations \(\,\mu \rightarrow \mu + \delta \mu \). In a narrow sense, it is a semi-group since the equation can only be integrated if one moves from small to large scales, but not the other way round.
- 25.
This effective action is not the same as the quantum action defined in Appendix D.2.2., which in the literature very often is also called effective action.
- 26.
And this without gauge bosons but at the cost of a coupling constant with mass dimension \(-\)2.
- 27.
Even Dirac, mentioned as the pioneer in higher-spin relativistic field equation, tried to derive these with algebraic acrobatics; there are no notions of symmetry.
- 28.
Interestingly, but rounding our understanding of symmetries, a gauge theory can also be obtained by enforcing constants of motion–originating from global symmetries in a theory–to constraints with Lagrangian multipliers [303].
- 29.
C. N. Yang: “If we were to rename them today, it is obvious that we should call the gauge invariance phase invariance, and the gauge fields should be called phase fields.” in “Geometry in physics”, in “To fulfill a vision–Jerusalem Einstein Centennial Symposium on Gauge Theories and Unification of Physical Forces,” (ed.) Y. Ne’eman, Addison-Wesley, New York, 1981.
- 30.
\(\mathbf {Spin(3,1)}\) is isomorphic to \(\mathbf {SL(2, \mathbb {C})}\).
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Sundermeyer, K. (2014). Relativistic Field Theory. In: Symmetries in Fundamental Physics. Fundamental Theories of Physics, vol 176. Springer, Cham. https://doi.org/10.1007/978-3-319-06581-6_5
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