By Mark Burgess
Publisher: Cambridge University Press
Print Publication Year: 2002
Online Publication Date:August 2009
Online ISBN:9780511535055
Hardback ISBN:9780521813631
Paperback ISBN:9780521675772
Chapter DOI: http://dx.doi.org/10.1017/CBO9780511535055.015
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In some branches of physics, such as condensed matter and quantum optics, one deals exclusively with non-relativistic models. However, there are occasionally advantages to using a relativistic formulation in quantum theory; by embedding a theory in a larger framework, one often obtains new insights. It is therefore useful to be able to take the non-relativistic limit of generally covariant theories, both as an indication of how large or small relativistic effects are and as a cultural bridge between covariant physics and non-relativistic quantum theory.
Particles and anti-particles
There is no unified theory of particles and anti-particles in the non-relativistic field theory. Formally there are two separate theories. When we take the non-relativistic limit of a relativistic theory, it splits into two disjoint theories: one for particles, with only positive definite energies, and one for anti-particles, with only negative definite energies. Thus, a non-relativistic theory cannot describe the interaction between matter and anti-matter.
The Green functions and fields reflect this feature. The positive frequency Wightman function goes into the positive energy particle theory, while the negative frequency Wightman function goes into the negative energy anti-particle theory. The objects which one then refers to as the Wightman functions of the non-relativistic field theory are asymmetrical. In normal Schrödinger field theory for matter, one says that the zero temperature negative frequency Wightman function is zero.
Klein–Gordon field
The free scalar field
We begin by considering the Klein–Gordon action for a real scalar field, since this is the simplest of the cases and can be treated at the level of the action. It also reveals several subtleties in the way quantities are defined and the names various quantities go by.
pp. i-vi
pp. vii-xviii
pp. xix-xx
Miscellaneous Frontmatter: Read PDF
pp. xxi-xxii
pp. 1-2
pp. 3-8
2 - The electromagnetic field: Read PDF
pp. 9-31
3 - Field parameters: Read PDF
pp. 32-49
4 - The action principle: Read PDF
pp. 50-71
5 - Classical field dynamics: Read PDF
pp. 72-112
6 - Statistical interpretation of the field: Read PDF
pp. 113-130
7 - Examples and applications: Read PDF
pp. 131-166
Part 2 - Groups and fields: Read PDF
pp. 167-168
8 - Field transformations: Read PDF
pp. 169-206
9 - Spacetime transformations: Read PDF
pp. 207-255
10 - Kinematical and dynamical transformations: Read PDF
pp. 256-282
11 - Position and momentum: Read PDF
pp. 283-324
12 - Charge and current: Read PDF
pp. 325-339
13 - The non-relativistic limit: Read PDF
pp. 340-357
14 - Unified kinematics and dynamics: Read PDF
pp. 358-389
15 - Epilogue: quantum field theory: Read PDF
pp. 390-396
Part 3 - Reference: a compendium of fields: Read PDF
pp. 397-398
16 - Gallery of definitions: Read PDF
pp. 399-409
17 - The Schrödinger field: Read PDF
pp. 410-415
18 - The real Klein–Gordon field: Read PDF
pp. 416-424
19 - The complex Klein–Gordon field: Read PDF
pp. 425-429
20 - The Dirac field: Read PDF
pp. 430-451
21 - The Maxwell radiation field: Read PDF
pp. 452-463
22 - The massive Proca field: Read PDF
pp. 464-465
23 - Non-Abelian fields: Read PDF
pp. 466-485
24 - Chern–Simons theories: Read PDF
pp. 486-490
25 - Gravity as a field theory: Read PDF
pp. 491-498
pp. 499-500
Appendix A - Useful formulae: Read PDF
pp. 501-512
Appendix B - Recommended reading: Read PDF
pp. 513-514
pp. 515-520
pp. 521-529