BRST symmetry

The Becchi-Rouet-Stora-Tyutin symmetry , BRST symmetry for short , sometimes just Becchi-Rouet-Stora symmetry ( BRS symmetry ), according to Carlo Becchi , Alain Rouet , Raymond Stora and Igor Tyutin , is a symmetry in the Quantum field theory , which is still present when the calibration of the quantum field has already been determined and the quantum field is no longer symmetrical in terms of calibration. This is made possible by the fact that the BRST symmetry, in addition to the original calibration symmetry, takes into account the existence of the unphysical Faddejew-Popow ghosts created by the calibration and the symmetry is built on the basis of anti- commutating Graßmann numbers .

background

The description of nature by quantum field theory is based in the standard model of elementary particle physics on the Yang-Mills theory . The basic physical forces such as the electromagnetic interaction are closely linked to symmetries in the Yang-Mills theory. A symmetry in this sense does not denote a spatial symmetry, but in general that physics does not change if the parameters of the quantum fields are changed. The change in the quantum fields according to a certain scheme that physics leaves unchanged is called the symmetry operation.

The physical information about the quantum fields and their equations of motion are contained in the Lagrangian of theory. It is therefore a sufficient condition that the Lagrangian of the theory does not change under symmetry operations. If certain terms occur additionally in the Lagrangian due to the symmetry operation on a certain field, then they must be compensated again by symmetry operations on another field. One such symmetry operation is the gauge transformation of the quantum fields. These calibration transformations already exist in classical electrodynamics , since certain terms can be added and subtracted from the auxiliary variables " vector potential " and " electrical potential " without changing the physical magnetic fields and electrical fields , as they cancel each other out in the interaction. One of the statements of the Yang-Mills theory is that there is a so-called calibration field for every gauge symmetry that exists in nature; in the case of electrodynamics, these are the light quanta (photons) .

On the other hand, when describing quantum field theory within the framework of the path integral formalism, the calibration of the calibration fields is required. Otherwise an infinite number of physically identical states that differ only in their calibration would be integrated, so that the integral is no longer well-defined. If any calibration is specified, this leads to an additional contribution to the Lagrangian, due to which the Lagrangian can no longer be invariant under further calibration transformations. In addition, quantum fields appear out of nowhere in the Lagrangian. These fields are called Faddejew-Popow ghost fields or ghosts for short after their first description, Ludwig Faddejew and Wiktor Popow . These ghosts have the unphysical property of not obeying the spin statistics theorem , since they (like the Higgs boson) are spin -0 particles, but instead of the Bose-Einstein statistics they follow the Fermi-Dirac statistics .

Since the ghosts are not originally part of the theory, they are not taken into account by the gauge symmetry, so the gauge symmetry operation does not specify any condition for the transformation of the ghosts. The BRST symmetry takes into account the existence of the ghosts to the extent that the transformation of the matter fields and the calibration fields does not differ from the calibration transformation, but the contribution by the calibration fixation term, which would violate the symmetry, is exactly compensated by the contribution by the spirit fields.

Mathematical description

If the Lagrangian is a Yang-Mills theory, then the entire Lagrangian consists of a kinetic part that describes the equations of motion of the calibration fields and their interaction with themselves , the calibration fixation term and the spirit fields, and possibly a term for matter and interaction between the forces and the matter. The gauge symmetry for the gauge fields implies that the Lagrangian remains invariant under the replacement for the gauge fields . They are arbitrary functions of space and time, are the structural constants of the symmetry group. ${\displaystyle {\mathcal {L}}}$ ${\displaystyle {\mathcal {L}}_{\text{kin}}}$ ${\displaystyle {\mathcal {L}}_{\text{fix}}}$ ${\displaystyle {\mathcal {L}}_{\text{Geist}}}$ ${\displaystyle A_{\mu }^{a}\to {A'}_{\mu }^{a}=A_{\mu }^{a}+\partial _{\mu }\alpha ^{a}+f^{abc}\alpha ^{b}A_{\mu }^{c}}$ ${\displaystyle A}$ ${\displaystyle \alpha }$ ${\displaystyle f}$

In quantum electrodynamics , the generalization of classical electrodynamics by quantum field theory, the vector potential, the structure constants and the specified transformation are the classical gauge transformation of the vector potential , under which the electric and magnetic fields do not change. ${\displaystyle A}$ ${\displaystyle f=0}$

R _ξ calibration

The calibration fixation term in the calibrations, which, in the case of quantum electrodynamics, generalize the Lorenz calibration in the Maxwell equations of electrodynamics, is ${\displaystyle R_{\xi }}$

${\displaystyle {\mathcal {L}}_{\text{fix}}=-{\frac {1}{2\xi }}(\partial ^{\mu }A_{\mu }^{a})^{2}}$

with the calibration parameter . The quantum field theoretical equivalent of the Lorenz calibration, the Feynman calibration , results from setting . The ghost term is ${\displaystyle \xi }$ ${\displaystyle \xi =1}$

${\displaystyle {\mathcal {L}}_{\text{Geist}}=-(\partial ^{\mu }{\bar {c}}^{a})(\partial _{\mu }c^{a}-f^{abc}A_{\mu }^{b}c^{c})}$

with the spirit fields , the anti-spirit fields and the structural constants of the symmetry group . In particular it is for the circle group so that in electrodynamics the spirit fields decouple and make no contribution. If the gauge transformation given above is inserted into these terms, the Lagrangian is no longer gauge invariant. ${\displaystyle c}$ ${\displaystyle {\bar {c}}}$ ${\displaystyle f}$ ${\displaystyle f=0}$

The invariance under the BRST transformation results from replacing the gauge transformation with the infinitesimal transformations

${\displaystyle {\begin{aligned}A_{\mu }^{a}\to {A'}_{\mu }^{a}&=A_{\mu }^{a}+\theta (\partial _{\mu }c^{a}+gf^{abc}A_{\mu }^{b}c^{c})\\{\bar {c}}^{a}\to {{\bar {c}}'}^{a}&={\bar {c}}^{a}+\theta {\frac {1}{\xi }}\partial ^{\mu }A_{\mu }^{a}\\c^{a}\to {c'}^{a}&=c^{a}-{\frac {1}{2}}\theta f^{abc}c^{b}c^{c}\end{aligned}}}$

with the infinitesimal transformation parameter , where is a Graßmann number. For these it is especially true that , but is, because the spirit fields themselves are Graßmann-valued. Since the product of two Graßmann numbers is again a "normal" number, the BRST transformation for the calibration field is not Graßmann-valued overall; only the calibration parameter is replaced by. Without taking the fermionic matter fields into account, the replacement rule also applies to them, so that the freedom of calibration for matter fields, corresponding to the free choice of the phase in classical quantum mechanics, can also be expressed by the BRST symmetry. ${\displaystyle \theta }$ ${\displaystyle \theta }$ ${\displaystyle \theta c=-c\theta }$ ${\displaystyle \theta A=A\theta }$ ${\displaystyle \alpha }$ ${\displaystyle \theta c}$

The transformation behavior for spirit and anti-spirit fields is different, but in contrast to matter and antimatter, there is no physical connection between spirits and anti-spirits, so that this does not generate a contradiction to physics.

General case

In general , the so-called Nakanishi-Lautrup field (not to be confused with the B field of classical electrodynamics) can be introduced without restricting the calibrations to calibrations . The Lagrangian then reads ${\displaystyle R_{\xi }}$ ${\displaystyle B}$

${\displaystyle {\mathcal {L}}={\mathcal {L}}_{\text{kin}}+{\bar {c}}^{a}\int \mathrm {d} ^{4}y\,{\frac {\delta g^{a}(A',x)}{\delta \alpha ^{b}(y)}}{\Bigg |}_{\alpha =0}c^{b}(y)+B^{a}g^{a}+{\frac {\xi }{2}}B^{a}B^{a}}$

where stands for a general calibration condition and for the functional derivation . The calibrations result from in connection with the Euler-Lagrange equations for the Nakanishi-Lautrup field . From this equation it can be seen that the Nakanishi-Lautrup field is only an auxiliary field, since no equations of motion of its own exist for it. The BRST symmetry operations valid in the general case are, if different from those in the calibration ${\displaystyle g}$ ${\displaystyle \delta g/\delta \alpha }$ ${\displaystyle R_{\xi }}$ ${\displaystyle g^{a}=\partial ^{\mu }A_{\mu }^{a}}$ ${\displaystyle \partial {\mathcal {L}}/\partial B^{a}=0}$ ${\displaystyle R_{\xi }}$

${\displaystyle {\begin{aligned}{\bar {c}}^{a}\to {{\bar {c}}^{a}}'&={\bar {c}}^{a}-\theta B^{a}\\B^{a}\to {B^{a}}'&=B^{a}\end{aligned}}}$

In particular, the BRST symmetry operation is nilpotent . That means, for any functional, the field operator is ${\displaystyle F[A,B,c,{\bar {c}}]}$

${\displaystyle (F'-F)'-(F'-F)=0}$.

Defined with the BRST operator , this can be briefly described as ${\displaystyle F'-F=\theta sF}$ ${\displaystyle s}$

${\displaystyle ssF=0}$

express. In particular, for the operation of the BRST operator on the calibration condition:

${\displaystyle sg^{a}=\int \mathrm {d} ^{4}y\,{\frac {\delta g^{a}(A',x)}{\delta \alpha ^{b}(y)}}{\Bigg |}_{\alpha =0}c^{b}(y)}$

This allows the Lagrangian to be compact as a

${\displaystyle {\mathcal {L}}={\mathcal {L}}_{\text{kin}}+s\left({\bar {c}}^{a}g^{a}+{\tfrac {\xi }{2}}{\bar {c}}^{a}B^{a}\right)}$

to be written. Since the BRST operation for the calibration fields is identical to the normal calibration transformation, it applies automatically . The Lagrangian is therefore due to the nilpotent of , invariant under BRST-Operatonen . ${\displaystyle s{\mathcal {L}}_{\text{kin}}=0}$ ${\displaystyle s}$ ${\displaystyle s{\mathcal {L}}=0}$

Consequences of the BRST symmetry

The calibration of the calibration fields is shown in the picture of the BRST operator; it is of the form . A new calibration can be chosen by adding any term of the form to the Lagrangian. In particular, the physics must not change under a gauge transformation, so that the following applies to two physical states according to Schwinger's quantum action principle: ${\displaystyle s\Psi }$ ${\displaystyle s\delta \Psi }$ ${\displaystyle |\alpha \rangle ,|\beta \rangle }$

${\displaystyle 0=\delta \langle \alpha |\beta \rangle =\langle \alpha |\mathrm {i} s\delta \Psi |\beta \rangle }$

Furthermore, according to Noether's theorem , every continuous local symmetry of the effect , i.e. also under symmetries of the Lagrangian, is followed by a conservation quantity and with it a continuity equation . The charge operator of the BRST charge obtained depends on the choice of calibration condition. In the calibrations it is ${\displaystyle R_{\xi }}$

${\displaystyle Q_{\text{BRST}}=\int \mathrm {d} ^{3}x\,\left[B^{a}(\partial _{t}c^{a}+f^{abc}A_{0}^{b}c^{c})-(\partial _{t}B^{a})c^{a}+{\frac {1}{2}}f^{abc}(\partial _{t}{\bar {c}}^{a})c^{b}c^{c}\right]}$

and generally satisfies the equation

${\displaystyle \mathrm {i} sF={\begin{cases}QF-FQ&\quad F{\text{ bosonisch}}\\QF+FQ&\quad F{\text{ fermionisch}}\end{cases}}}$.

Therefore must for each physical condition

${\displaystyle \langle \alpha |QF|\beta \rangle \mp \langle \alpha |FQ|\beta \rangle }$ = 0

with apply from what ${\displaystyle F=\delta \Psi }$

${\displaystyle \langle \alpha |Q=Q|\beta \rangle =0}$

follows. If the BRST loading operator operates on a physical state, the result is zero. In other words, any physical state has no BRST charge; it is at the core of the BRST charge operator. Ghosts and anti-ghosts, on the other hand, have a BRST charge, so this is a different formulation for the fact that they do not describe any physical states.

Furthermore, it follows from the Nil power of that the square of the BRST charge operator must be either zero or the identity . Since the BRST charge operator does not have a vanishing ghost quantum number, must be. This means that two identical physical states can differ by a state vector of the form . ${\displaystyle s}$ ${\displaystyle Q^{2}=0}$ ${\displaystyle Q|\dots \rangle }$

literature

• Manfred Böhm, Ansgar Denner and Hans Joos: Gauge Theories of the Strong and Electroweak Interaction . 3. Edition. Teubner, Stuttgart Leipzig Wiesbaden 2001, ISBN 3-519-23045-3 (English).
• Mattew D. Schwartz: Quantum Field Theory and the Standard Model . Cambridge University Press, Cambridge 2014, ISBN 978-1-107-03473-0 (English).
• Steven Weinberg: The Quantum Theory of Fields Volume II: Modern Applications . Cambridge University Press, Cambridge 1996, ISBN 0-521-55002-5 (English).