Slavnov-Taylor identity

The Slavnow-Taylor identities are a class of relationships in quantum field theory that link vacuum expectation values of time-ordered quantities, the so-called correlation functions. They generalize the Ward-Takahashi identity of quantum electrodynamics to non-Abelian Yang-Mills theories , especially quantum chromodynamics , i.e. the theory of strong interaction . The Slavnov-Taylor identities were independently discovered by John C. Taylor in 1971 and Andrei Slavnov in 1972 .

The Slavnow-Taylor identities follow from the BRST symmetry , since physics must be invariant under BRST symmetry operations. This means that a modification of all fields must not change the physics. This applies in particular to the generating functional ${\ displaystyle \ Phi \ to \ Phi '= \ Phi + \ delta \ Phi}$

{\ displaystyle {\ mathcal {Z}} [J] = \ int \ prod _ {i = 1} ^ {n} {\ mathcal {D}} [\ Phi _ {i}] \ exp \ left (\ mathrm {i} \ int \ mathrm {d} ^ {4} x \ left ({\ mathcal {L}} + \ sum _ {j = 1} ^ {n} J_ {j} (x) \ Phi _ {j } (x) \ right) \ right)}

in the path integral formalism of quantum field theory. That is, it applies

{\ displaystyle \ delta {\ mathcal {Z}} = {\ mathcal {Z}} '- {\ mathcal {Z}} = 0}

.

Since the Lagrangian and the path integral are already invariant under BRST operations, so it is and , and the BRST symmetry is based on nilpotent Graßmann numbers , it follows from this ${\ displaystyle {\ mathcal {L}}}$ ${\ displaystyle {\ mathcal {L}} '= {\ mathcal {L}}}$ ${\ displaystyle {\ mathcal {D}} \ Phi '= {\ mathcal {D}} \ Phi}$

{\ displaystyle \ int \ prod {\ mathcal {D}} \ Phi _ {i} \ exp \ left ({\ mathcal {i}} \ int \ mathrm {d} ^ {4} x \ left ({\ mathcal {L}} + \ sum J_ {j} (x) \ Phi _ {j} (x) \ right) \ right) \ mathrm {i} \ int \ mathrm {d} ^ {4} yJ_ {i} ( y) \ delta \ Phi _ {i} (y) = 0}

The time-ordered vacuum expectation values are now obtained by differentiating according to the corresponding values and then setting all of them to zero. Since there is always exactly one left with this procedure , it follows ${\ displaystyle J}$ ${\ displaystyle J}$ ${\ displaystyle \ delta \ Phi}$

{\ displaystyle {\ begin {aligned} 0 & = \ langle T (\ delta \ Phi _ {1}) \ Phi _ {2} \ dots \ Phi _ {n} \ rangle + \ dots + \ langle T \ Phi _ {1} \ dots \ Phi _ {n-1} (\ delta \ Phi _ {n}) \ rangle \\ & = \ delta \ langle T \ Phi _ {1} \ dots \ Phi _ {n} \ rangle \ end {aligned}}}

This is the Slavnov-Taylor identity.

Individual evidence

↑ John C. Taylor: Ward identities and charge renormalization of the Yang-Mills field . In: Nucl. Phys. B . tape 33 , no. 2 , 1971, p. 436-444 , doi : 10.1016 / 0550-3213 (71) 90297-5 (English).
^ Andrei Slavnov: Ward identities in gauge theories . In: Theoretical and Mathematical Physics . tape 10 , no. 2 , 1972, p. 99-104 , doi : 10.1007 / BF01090719 (English).

[1] John C. Taylor: Ward identities and charge renormalization of the Yang-Mills field . In: Nucl. Phys. B . tape 33 , no. 2 , 1971, p. 436-444 , doi : 10.1016 / 0550-3213 (71) 90297-5 (English).

[2] Andrei Slavnov: Ward identities in gauge theories . In: Theoretical and Mathematical Physics . tape 10 , no. 2 , 1972, p. 99-104 , doi : 10.1007 / BF01090719 (English).