Abstract
The major theme of quantum field theory is the development of a unified theory that may be used to describe nature from microscopic to cosmological distances. Quantum field theory was born 90 years ago, when quantum theory met relativity, and has captured the hearts of the brightest theoretical physicists in the world. It is still going strong. It has gone through various stages, met various obstacles on the way, and has been struggling to provide us with a coherent description of nature in spite of the “patchwork” of seemingly different approaches that have appeared during the last 40 years or so, but still all, with the common goal of unification.
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Notes
- 1.
- 2.
For intricacies of Wigner’s Symmetry Transformations in quantum mechanics, see, e.g., Manoukian [1], pp. 55–65.
- 3.
More generally one may also have initially a mixture described by a density operator.
- 4.
One should also consider unit rays in this discussion and in the subsequent definitions but we will not go into these points here.
- 5.
One may conveniently, in general, absorb δξ in G.
- 6.
With the Minkowski metric adopted in this book [ η μ ν ] = diag[−1, 1, 1, 1], b i = b i , b 0 = −b 0.
- 7.
One may also introduce more than one such operator.
- 8.
This will be studied in the accompanying book Manoukian [2]: Quantum Field Theory II: Introductions to Quantum Gravity, Supersymmetry, and string Theory. Minkowski spacetime is, in turn, extended to what has been called superspace.
- 9.
The reader may wish to consult Manoukian [1], pp. 55–65, 112–115, where additional details, and proofs, are spelled out in quantum theory.
- 10.
Note that the factors \((\rho _{j}\! -\!\alpha _{j})(\overline{\rho }_{j}\! -\!\overline{\alpha }_{j})\) commute with everything.
- 11.
We interchangeably use the notations δ(x − x ′) and δ (D )(x − x ′) where D is the dimensionality of spacetime.
- 12.
See Problem 2.8.
References
Manoukian, E. B. (2006). Quantum theory: A wide spectrum. Dordrecht: Springer.
Manoukian, E. B. (2016). Quantum field theory II: Introductions to quantum gravity, supersymmetry, and string theory. Dordrecht: Springer.
Wigner, E. P. (1959). Group theory, and its applications to the quantum mechanics of atomic spectra. New York: Academic.
Wigner, E. P. (1963). Invariant quantum mechanical equations of motion. In: Theoretical physics (p. 64). Vienna: International Atomic Energy Agency.
Recommended Reading
Manoukian, E. B. (2006). Quantum theory: A wide spectrum. Dordrecht: Springer.
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Manoukian, E.B. (2016). Preliminaries. In: Quantum Field Theory I. Graduate Texts in Physics. Springer, Cham. https://doi.org/10.1007/978-3-319-30939-2_2
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DOI: https://doi.org/10.1007/978-3-319-30939-2_2
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