Schwinger's quantum action principle: Difference between revisions
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{{distinguish|text=Schwinger's variational principle, ''[[Schwinger variational principle]]''}} |
{{distinguish|text=Schwinger's variational principle, ''[[Schwinger variational principle]]''}} |
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The '''Schwinger's quantum action principle''' is a [[variational method|variational]] approach to [[quantum mechanics]] and [[quantum field theory]].<ref>{{Cite book |last=Schwinger |first=Julian |url=http://link.springer.com/10.1007/978-3-662-04589-3 |title=Quantum Mechanics |date=2001 |publisher=Springer Berlin Heidelberg |isbn=978-3-642-07467-7 |editor-last=Englert |editor-first=Berthold-Georg |location=Berlin, Heidelberg |language=en |doi=10.1007/978-3-662-04589-3}}</ref><ref>{{Citation |last=Dittrich |first=Walter |title=The Quantum Action Principle |date=2021 |url=https://link.springer.com/10.1007/978-3-030-69105-9_11 |work=The Development of the Action Principle |pages=79–82 |place=Cham |publisher=Springer International Publishing |language=en |doi=10.1007/978-3-030-69105-9_11 |isbn=978-3-030-69104-2 |access-date=2022-10-19}}</ref> This theory was introduced by [[Julian Schwinger]] in a series of articles starting 1950.<ref>{{Cite journal |last=Schweber |first=Silvan S. |date=2005-05-31 |title=The sources of Schwinger's Green's functions |url=https://pnas.org/doi/full/10.1073/pnas.0405167101 |journal=Proceedings of the National Academy of Sciences |language=en |volume=102 |issue=22 |pages=7783–7788 |doi=10.1073/pnas.0405167101 |issn=0027-8424 |pmc=PMC1142349 |pmid=15930139}}</ref> |
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the two formalisms are identical. |
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== Approach == |
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| ⚫ | In Schwingers approach, the [[Action (physics)|action principle]] is targeted towards quantum mechanics. The action becomes a '''quantum action''', i.e. an operator, <math> S </math>. Although it is superficially different from the [[path integral formulation]] where the action is a classical function, the modern formulation of the two formalisms are identical. |
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Suppose we have two states defined by the values of a [[complete set of commuting operators]] at two times. Let the early and late states be <math>| A \rang</math> and <math>| B \rang</math>, respectively. Suppose that there is a parameter in the Lagrangian which can be varied, usually a source for a field. The main equation of '''Schwinger's quantum action principle''' is: |
Suppose we have two states defined by the values of a [[complete set of commuting operators]] at two times. Let the early and late states be <math>| A \rang</math> and <math>| B \rang</math>, respectively. Suppose that there is a parameter in the Lagrangian which can be varied, usually a source for a field. The main equation of '''Schwinger's quantum action principle''' is: |
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:<math> \delta \langle B|A\rangle = i \langle B| \delta S |A\rangle,\ </math> |
:<math> \delta \langle B|A\rangle = i \langle B| \delta S |A\rangle,\ </math> |
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where the derivative is with respect to small changes in the parameter. |
where the derivative is with respect to small changes (<math>\delta</math>) in the parameter, and <math>S=\int \mathcal{L} \, \mathrm{d}t</math> with <math>\mathcal{L}</math> the [[Lagrangian mechanics|Lagrange]] operator. |
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In the path integral formulation, the transition amplitude is represented by the sum |
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| ⚫ | over all histories of <math>\exp(iS)</math>, with appropriate [[boundary condition]]s representing the states <math>| A \rang</math> and <math>| B \rang</math>. The infinitesimal change in the amplitude is clearly given by Schwinger's formula. Conversely, starting from Schwinger's formula, it is easy to show that the fields obey canonical commutation relations and the classical equations |
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of motion, and so have a path integral representation. Schwinger's formulation was most significant because it could treat fermionic anticommuting fields with the same formalism as bose fields, thus implicitly introducing differentiation and integration |
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with respect to anti-commuting coordinates. |
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| ⚫ | In the path integral formulation, the transition amplitude is represented by the sum over all histories of <math>\exp(iS)</math>, with appropriate [[boundary condition]]s representing the states <math>| A \rang</math> and <math>| B \rang</math>. The infinitesimal change in the amplitude is clearly given by Schwinger's formula. Conversely, starting from Schwinger's formula, it is easy to show that the fields obey canonical commutation relations and the classical equations of motion, and so have a path integral representation. Schwinger's formulation was most significant because it could treat fermionic anticommuting fields with the same formalism as bose fields, thus implicitly introducing differentiation and integration with respect to anti-commuting coordinates. |
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==External links== |
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* [http://www.pnas.org/cgi/content/full/102/22/7783 A brief (but very technical) description of Schwinger's paper] |
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== References == |
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<references /> |
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[[Category:Perturbation theory]] |
[[Category:Perturbation theory]] |
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[[Category:Quantum field theory]] |
[[Category:Quantum field theory]] |
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Revision as of 12:30, 19 October 2022
The Schwinger's quantum action principle is a variational approach to quantum mechanics and quantum field theory.[1][2] This theory was introduced by Julian Schwinger in a series of articles starting 1950.[3]
Approach
In Schwingers approach, the action principle is targeted towards quantum mechanics. The action becomes a quantum action, i.e. an operator, . Although it is superficially different from the path integral formulation where the action is a classical function, the modern formulation of the two formalisms are identical.
Suppose we have two states defined by the values of a complete set of commuting operators at two times. Let the early and late states be and , respectively. Suppose that there is a parameter in the Lagrangian which can be varied, usually a source for a field. The main equation of Schwinger's quantum action principle is:
where the derivative is with respect to small changes () in the parameter, and with the Lagrange operator.
In the path integral formulation, the transition amplitude is represented by the sum over all histories of , with appropriate boundary conditions representing the states and . The infinitesimal change in the amplitude is clearly given by Schwinger's formula. Conversely, starting from Schwinger's formula, it is easy to show that the fields obey canonical commutation relations and the classical equations of motion, and so have a path integral representation. Schwinger's formulation was most significant because it could treat fermionic anticommuting fields with the same formalism as bose fields, thus implicitly introducing differentiation and integration with respect to anti-commuting coordinates.
References
- ^ Schwinger, Julian (2001). Englert, Berthold-Georg (ed.). Quantum Mechanics. Berlin, Heidelberg: Springer Berlin Heidelberg. doi:10.1007/978-3-662-04589-3. ISBN 978-3-642-07467-7.
- ^ Dittrich, Walter (2021), "The Quantum Action Principle", The Development of the Action Principle, Cham: Springer International Publishing, pp. 79–82, doi:10.1007/978-3-030-69105-9_11, ISBN 978-3-030-69104-2, retrieved 2022-10-19
- ^ Schweber, Silvan S. (2005-05-31). "The sources of Schwinger's Green's functions". Proceedings of the National Academy of Sciences. 102 (22): 7783–7788. doi:10.1073/pnas.0405167101. ISSN 0027-8424. PMC 1142349. PMID 15930139.
{{cite journal}}: CS1 maint: PMC format (link)