Bosonic field: Difference between revisions
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In [[quantum field theory]], a '''bosonic field''' is a [[quantum field]] whose quanta are [[boson]]s; that is, they obey [[Bose-Einstein statistics]]. Bosonic fields obey [[canonical commutation relation]]s, as distinct from the [[canonical anticommutation relation]]s obeyed by [[fermionic fields]]. |
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Examples include scalar fields, describing spin 0 particles such as the [[Higgs boson]], and gauge fields, describing spin 1 particles such as the photon. |
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==Basic properties== |
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Free (non-interacting) bosonic fields obey [[canonical commutation relation]]s. Those relations also hold for interacting bosonic fields in the interaction picture, where the fields evolve in time as if free and the effects of the interaction are encoded in the evolution of the states. It is these commutation relations that imply Bose-Einstein statistics for the field quanta. |
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==Examples== |
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Examples of bosonic fields include [[scalar field (physics)|scalar field]]s, [[gauge field]]s, and [[symmetric tensor|symmetric 2-tensor]] [[tensor field|fields]], which are characterized by their [[covariance]] under [[Lorentz transformation]]s and have spins 0, 1 and 2, respectively. Physical examples, in the same order, are the Higgs field, the photon field, and the graviton field. While the first one remains to be observed, it is widely believed to exist. Of the last two, only the photon field can be quantized using the conventional methods of canonical or path integral quantization. This has led to the theory of quantum electrodynamics, one of the most successful theories in physics. [[Quantum gravity|Quantization of gravity]], on the other hand, is a long standing problem that has led to development of theories such as [[string theory]] and [[loop quantum gravity]]. |
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==Spin and statistics== |
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The [[spin-statistics theorem]] implies that quantization of local, relativistic field theories in 3+1 dimensions may lead either to bosonic or fermionic quantum fields, i.e., fields obeying commutation or anti-commutation relations, according to whether they have [[integer]] or [[half integer]] spin, respectively. Thus bosonic fields are one of the two theoretically possible types of quantum field, namely those corresponding to particles with integer spin. |
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In a non-relativistic many-body theory, the spin and the statistical properties of the quanta are not directly related. In fact, the commutation or anti-commutation relations are assumed based on whether the theory one intends to study corresponds to particles obeying Bose-Einstein or Fermi-Dirac statistics. In this context the spin remains an internal quantum number that is only phenomenologically related to the statistical properties of the quanta. Examples of non-relativistic bosonic fields include those describing cold bosonic atoms, such as Helium-4. |
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Such non-relativistic fields are not as fundamental as their relativistic counterparts: they provide a convenient 're-packaging' of the many-body wave function describing the state of the system, whereas the relativistic fields described above are a necessary consequence of the consistent union of relativity and quantum mechanics. |
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==See also== |
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*[[Fermionic field]] |
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*[[Spin-statistics theorem]] |
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*[[Quantum triviality]] |
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==References== |
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* Edwards, D. (1981). ''The Mathematical Foundations of Quantum Field Theory: Fermions, Gauge Fields, and Super-symmetry, Part I: Lattice Field Theories,'' International J. of Theor. Phys., Vol. 20, No. 7. |
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* Hoffmann, S.E. et alia (2008) 'Hybrid Phase-Space Simulation Method for Interacting Bose Fields'. ''Physical Review A'' Vol. 78, Issue 1. |
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* Peskin, M and Schroeder, D. (1995). ''An Introduction to Quantum Field Theory,'' Westview Press. |
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* Srednicki, Mark (2007). ''[http://www.physics.ucsb.edu/~mark/qft.html Quantum Field Theory]'', Cambridge University Press, ISBN 978-0521864497. |
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* Weinberg, Steven (1995). ''The Quantum Theory of Fields,'' (3 volumes) Cambridge University Press. |
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{{physics-stub}} |
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[[Category:Quantum field theory]] |
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