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The Problem of Time pp 65–78Cite as

Quantum Field Theory (QFT)

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Part of the book series: Fundamental Theories of Physics ((FTPH,volume 190))

Abstract

This chapter outlines (special-relativistic) quantum field theory, with particular emphasis on the roles played therein by time and background dependence. (Fully understanding this chapter rather probably requires having studied at least an elementary course in quantum field theory.) In particular, the current chapter shows that the Wightman axioms for quantum field theory are replete with temporal and background dependent concepts. The Lorentz group, upon which quantum field theory’s equations for each spin of particle rest, is also a background-dependent concept.

While there is some interesting contrast with the previous chapter’s account of time in nonrelativistic quantum theory, the main point of this chapter’s exposition is the differences between it and how time is conceived of in general relativity (Chaps. 7 and 8). These differences are essential preliminary reading before tackling both Chapter 11’s introduction to Quantum Gravity and Chaps. 9, 10 and 12’s introduction to this book’s main topic of the Problem of Time and its Background Independence underpinning. The current chapter also contains this book’s quantum exercises.

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Notes

  1. 1.

    If unfamiliar with any of the non-temporal material in this Chapter, consult [712]; some of this Chapter’s material on time in QFT (and the Wightman axioms) may however be new to the reader.

  2. 2.

    If insufficiently familiar with Green’s functions (named after 19th century mathematician George Green), consult [220]. In contrast to propagators, correlators are Green’s functions corresponding to time-independent wave equations. Finally, examples of choices of contour used in defining propagators are retarded, advanced, Feynman and ‘anti-Feynman’ propagators, after noted physicist Richard Feynman. The retarded case is causal and the advanced case is anti-causal; The most usual choice for propagators is the Feynman propagator, which is the half-causal half-anticausal choice, with matter treated causally and antimatter treated anti-causally.

  3. 3.

    The capital Latin indices here run over 1 to 4 for 4-component spinor indices; these spinorial indices are often suppressed in this book.

  4. 4.

    By the half-causal half-anticausal choice in the Feynman propagators, some of the arrows in Feynman diagrams (Fig. 6.1.b) point backwards. This corresponds to modelling distinguishable antiparticles as if they were travelling backward in time. This convention moreover indeed works for practical purposes.

  5. 5.

    In this book, we use overhead arrows to denote 4-vector quantities.

  6. 6.

    QFT does not by itself entail SR, e.g. phonons are a Nonrelativistic QFT model of quanta of sound waves in materials.

  7. 7.

    Moreover, the Wightman axioms (named after mathematical physicist Arthur Wightman) do not cover Quantum Gauge Theories. See e.g. [269] for a recent overview of this, and also [687] as regards an advanced consideration of their Euclidean counterpart.

  8. 8.

    Appendix O outlines ‘distributions’ in the current sense of Functional Analysis. So as to not confuse this with ‘probability distribution’ or ‘distribution of matter’, this book always fully spells out these other uses.

  9. 9.

    I.e. physicist Murray Gell-Mann’s eightfold way explanation of the octet, singlet and decuplet patterns of the observed and predicted-and-confirmed hadrons [886].

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  1. DAMTP, Centre for Mathematical Sciences, Cambridge, UK

    Edward Anderson

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Anderson, E. (2017). Quantum Field Theory (QFT). In: The Problem of Time. Fundamental Theories of Physics, vol 190. Springer, Cham. https://doi.org/10.1007/978-3-319-58848-3_6

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