Gell-Mann–Low theorem
E59631
The Gell-Mann–Low theorem is a fundamental result in quantum field theory that rigorously connects interacting quantum fields to free fields via the adiabatic switching-on of interactions, underpinning the use of perturbation theory and the Dyson series.
Statements (47)
| Predicate | Object |
|---|---|
| instanceOf |
theorem in quantum field theory
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| appliesTo |
Heisenberg picture fields
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interaction picture fields → |
| assumes |
existence of adiabatic limit
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stability of the vacuum under adiabatic switching → unique interacting vacuum state → |
| category |
theorem in mathematical physics
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| clarifies |
relation between bare and interacting fields
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role of interaction picture in QFT → |
| concerns |
time evolution with switched-on interaction Hamiltonian
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vacuum expectation values of time-ordered products → |
| connects |
time-ordered correlation functions of interacting fields to free-field ones
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| context |
renormalized perturbation theory
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|
| field |
quantum field theory
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|
| formalism |
operator formalism of quantum field theory
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| hasConsequence |
expression of interacting Green’s functions via functional derivatives of generating functional
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justification of perturbative expansion around free theory → |
| historicalContext |
developed in early years of renormalized quantum field theory
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| implies |
Dyson series expansion for the S-matrix
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| importance |
fundamental result for the foundations of perturbative QFT
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| involves |
S-matrix
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time-ordered exponentials → vacuum-to-vacuum transition amplitudes → |
| mathematicalFormulation |
expresses interacting vacuum as limit of time-evolution operator acting on free vacuum
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| namedAfter |
Francis E. Low
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Murray Gell-Mann → |
| provides |
rigorous connection between interacting and free fields
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| relatedTo |
Dyson’s formula
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LSZ reduction formula → interaction picture time-evolution operator → renormalization theory → |
| relates |
free (bare) vacuum state
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free quantum fields → interacting quantum fields → interacting vacuum state → |
| requires |
adiabatic factor e^{-\epsilon |t|} in interaction Hamiltonian
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| underpins |
Dyson series
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perturbation theory in quantum field theory → |
| usedBy |
quantum field theorists
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| usedFor |
calculation of correlation functions
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construction of perturbative expansions → definition of Green’s functions → derivation of Feynman rules → |
| usedIn |
perturbative calculations in particle physics
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relativistic quantum field theory → scattering theory → |
| usesConcept |
adiabatic switching of interactions
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|
Referenced by (1)
| Subject (surface form when different) | Predicate |
|---|---|
|
Dyson series
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relatedTo |