Abstract
This chapter describes the quantization of field theories on curved spacetimes, highlighting the differences between quantum field theory on Minkowski space. Beginning with the action describing a real scalar field, the equations of motion for the field are derived and an appropriate inner product on the space of solutions is introduced. The canonical quantization of this field theory is carried out, emphasizing the non-uniqueness of the vacuum state. The particle interpretation of such a field theory is described in detail with an emphasis on the importance of an operational definition of particles.
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Notes
- 1.
In this chapter we will confine the solutions u i(x) to a (n − 1)-tours of side length L, i.e. box normalized solutions. This results in the solutions being labelled by discrete indices. To convert to the continuum normalization one should replace \(\left (\frac {2 \pi }{L}\right )^{n-1}\sum _i\) with \(\int dk^{n-1}\) and the Kronecker delta with the Dirac delta function.
- 2.
The use of the word “particle” here is different than how the word is used in everyday language. The word particle commonly refers to an object with a well-defined energy, momentum, and position. Here, the use of the word particle refers to excitations of the field that have a well-defined energy. However, these particles are global excitations of the field and therefore do not have a well-defined position.
References
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S.A. Fulling, Nonuniqueness of canonical field quantization in Riemannian space-time. Phys. Rev. D 7, 2850 (1973)
V.F. Mukhanov, S. Winitzki, Introduction to Quantum Effects in Gravity (Cambridge University Press, Cambridge, 2007)
W.G. Unruh, Notes on blackhole evaporation. Phys. Rev. D 14, 870 (1976)
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Smith, A.R.H. (2019). Quantum Field Theory on Curved Spacetimes. In: Detectors, Reference Frames, and Time. Springer Theses. Springer, Cham. https://doi.org/10.1007/978-3-030-11000-0_2
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DOI: https://doi.org/10.1007/978-3-030-11000-0_2
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