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BRST Quantization of Yang-Mills Theory: a Purely Hamiltonian Approach on Fock Space

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BRST Quantization of Yang-Mills Theory: a Purely Hamiltonian Approach on Fock Space PHYSICAL REVIEW D 97, 074006 (2018) BRST quantization of Yang-Mills theory: A purely Hamiltonian approach on Fock space Hans Christian Öttinger* ETH Zürich, Department of Materials, Polymer Physics, HCP F 47.2, CH-8093 Zürich, Switzerland (Received 1 March 2018; published 3 April 2018) We develop the basic ideas and equations for the BRST quantization of Yang-Mills theories in an explicit Hamiltonian approach, without any reference to the Lagrangian approach at any stage of the development. We present a new representation of ghost fields that combines desirable self-adjointness properties with canonical anticommutation relations for ghost creation and annihilation operators, thus enabling us to characterize the physical states on a well-defined Fock space. The Hamiltonian is constructed by piecing together simple BRST invariant operators to obtain a minimal invariant extension of the free theory. It is verified that the evolution equations implied by the resulting minimal Hamiltonian provide a quantum version of the classical Yang-Mills equations. The modifications and requirements for the inclusion of matter are discussed in detail. DOI: 10.1103/PhysRevD.97.074006 I. INTRODUCTION The purpose of this paper is to show in the context of BRST quantization is a pivotal tool in developing Yang-Mills theories how all the above facets can be theories of the fundamental interactions, where the acro- handled entirely within the Hamiltonian approach, where nym BRST refers to Becchi, Rouet, Stora [1] and Tyutin explicit constructions on a suitable Fock space allow for a [2]. This method for handling constraints in the quantiza- maximum of intuition. The focus on Fock space implies a (quantum) particle interpretation rather than a field tion of field theories usually requires a broad viewpoint idealization. The signed and canonical inner products are because it covers a number of important aspects. The particularly transparent on Fock space. By including constraints are related to gauge symmetry, which suggests temporal and longitudinal in addition to transverse gauge that a Lagrangian approach is preferable, in particular, bosons (we typically think of photons or gluons), Lorentz as also Lorentz symmetry needs to be incorporated. By symmetry is enabled at the early stage of constructing the Noether’s theorem, symmetries come with conserved underlying Fock space. All symmetry arguments are based quantities, which suggest to focus on time evolution and on the BRST charge, the construction of which relies on its hence to favor the Hamiltonian approach. Practical calcu- role as the generator of BRST transformations, the quantum lations, for example in perturbation theory, are most version of gauge transformations. Of course, the BRST conveniently done in terms of the path-integral formulation charge has to respect also Lorentz symmetry. and hence on the Lagrangian side. The identification of In this paper, we present a new way of introducing ghost physical states, which requires gauge fixing as an addi- particles. Whereas one usually has to make the choice tional aspect of introducing constraints, is most naturally between natural anticommutation relations for the ghost done on Hilbert space (as the space of all states). The issue creation and annihilation operators on the one hand (see, of signed inner products, to be considered simultaneously e.g., [3]) and self-adjointness of the BRST charge on the with canonical inner products, requires particular attention other hand (see, e.g., [4]), we here propose a representation in constructing the physical states and should clearly of ghost particles that combines both properties. This is a benefit from a simple and intuitive approach to BRST crucial advantage because “In the non-Abelian case, the quantization. removal of unphysical gauge boson polarizations is more subtle [than in the Abelian case], and we have seen that it involves the ghosts in an essential way” (see p. 520 of [5]). *[email protected]; www.polyphys.mat.ethz.ch We simultaneously have a well-defined Fock space and the powerful tools required to select the physical states of a Published by the American Physical Society under the terms of gauge theory in the BRST approach. the Creative Commons Attribution 4.0 International license. Further distribution of this work must maintain attribution to After constructing a number of simple BRST invariant the author(s) and the published article’s title, journal citation, operators, these operators can be used to build up the and DOI. Funded by SCOAP3. BRST invariant Hamiltonian. By piecing together invariant 2470-0010=2018=97(7)=074006(14) 074006-1 Published by the American Physical Society HANS CHRISTIAN ÖTTINGER PHYS. REV. D 97, 074006 (2018) μ a abc bμ c μ a abc bμ c operators to reproduce the proper Hamiltonian of the free ∂ Fμν − gf A Fμν → ∂ Fμν − gf A Fμν theory for vanishing interaction strength, one obtains the abc μ b bde dμ e c 2 − gf ð∂ Fμν − gf A FμνÞΛ þ OðΛ Þ; ð6Þ Hamiltonian of Yang-Mills theory as a minimal BRST invariant extension of the free theory. The validity of the Hamiltonian can be verified by comparing the time we realize that the field equations (2) in the absence of evolution implied by this Hamiltonian to the classical external currents are gauge invariant. Moreover, the gauge evolution equations. Matter can be included into the transformation behavior of external currents required to Hamiltonian approach with the help of the current algebra. obtain gauge invariant field equations becomes evident, a a abc b c Jμ → Jμ − gf JμΛ : ð7Þ II. CLASSICAL YANG-MILLS THEORY a Yang-Mills theory introduces antisymmetric fields Fμν For the gauge bosons, we here impose the covariant Lorenz a that are defined in terms of four-vector potentials Aμ, gauge condition aμ a a a abc b c ∂μA 0: 8 Fμν ¼ ∂μAν − ∂νAμ − gf AμAν; ð1Þ ¼ ð Þ where the superscripts a, b, c label the generators of the We finally rewrite the Yang-Mills equations in a par- underlying symmetry group and the indices μ, ν label the ticular inertial system in the Maxwellian form which, in space-time components; the parameter g is the strength of contrast to all the above equations, is no longer manifestly the interaction and the set of numbers fabc are the structure Lorentz covariant. Such a reformulation is a straightfor- constants of the underlying Lie group. The field equations ward exercise (see, e.g., [7,8] with some deviations in the are given by choice of signs). The counterparts of electric and magnetic fields are μ a abc bμ c a obtained by the convention ∂ Fμν − gf A Fμν ¼ −Jν ; ð2Þ 0 1 0 − a − a − a a E1 E2 E3 where the four-vector Jν is an external current. The B C Ea 0 Ba −Ba parameter g usually occurs with the opposite sign because a B 1 3 2 C Fμν B C: 9 − ð Þ¼@ a a a A ð Þ we here choose the signature ð ; þ; þ; þÞ for the E2 −B3 0 B1 Minkowski metric, contrary to the more common con- a a − a 0 aμ E3 B2 B1 vention ðþ; −; −; −Þ; the four-vectors ∂μ and A are independent of the signature of the Minkowski metric. As pointed out before, for the non-Abelian case, these ˜ a − abc bμ c The term Jν ¼ gf A Fμν may be interpreted as the fields are not gauge invariant, current carried by the gauge bosons. The structure con- stants can be assumed to be completely antisymmetric in Ba → Ba − gfabcBbΛc þ OðΛ2Þ; ð10Þ the indices a, b, c (see Sec. 15.4 of [5]). They moreover satisfy the Jacobi identity and ads bcs bds cas cds abs 0 f f þ f f þ f f ¼ : ð3Þ Ea → Ea − gfabcEbΛc þ OðΛ2Þ: ð11Þ This identity is repeatedly needed in analyzing the gauge For convenience, we define the additional component transformation behavior of the classical and quantum Yang- Mills equations. a aμ E ¼ ∂μA ; ð12Þ Let us consider gauge transformations, which are given 0 by (see, e.g., pp. 490f of [5] or Sec. 15.1 of [6]) so that the Lorenz gauge condition (8) can be expressed as Ea ¼ 0. a → a ∂ Λa − abc bΛc 0 Aμ Aμ þ μ gf Aμ : ð4Þ The gauge-dependent field definitions (1) become Unlike for the Abelian case, the resulting transformation 1 Ba ∇ Aa − abcAb Ac ¼ × 2 gf × ; ð13Þ a a abc b c 2 Fμν → Fμν − gf FμνΛ þ OðΛ Þ; ð5Þ ∂Aa Ea ¼ ∇Aa − − gfabcAbAc; ð14Þ implies that the fields are not gauge invariant for non- 0 ∂t 0 Abelian Yang-Mills theories. However, by considering the combined transformation law and the field equations (2) can be written as 074006-2 BRST QUANTIZATION OF YANG-MILLS THEORY: A … PHYS. REV. D 97, 074006 (2018) ∇ Ea − abcAb Ec − a a0 ∂2 · gf · ¼ J0 ¼ J ; ð15Þ 2 a a abc bμ c c − ∇ A ¼ J þ gf A ð∇Aμ − 2∂μA Þ ∂t2 2 abs cds bμ c d ∂Ea þ g f f A AμA − ∇ × Ba − gfabc AbEc − Ab × Bc −Ja: 16 ð 0 Þ¼ ð Þ ac abc b cμ ∂t − ðδ ∇ − gf A Þ∂μA ; ð23Þ Equations (13)–(16) correspond to Eqs. (3.15), (4.2c), in each of which the last term disappears for the Lorenz (4.2a) and (4.2b) of [7]. In Eqs. (15) and (16), we recognize gauge (8) (cf. Appendix of [8]). A closer look at Eqs. (22) the temporal and spatial components of the current and (23) reveals that one recovers Lorentz covariance in abc bμ c these wave equations. −gf A Fμν associated with the fields, III. FOCK SPACE J˜ a0 ¼ gfabcAb · Ec; ð17Þ We introduce a Fock space together with a collection of aα† aα adjoint operators aq and aq creating and annihilating ˜ a abc b c b c J ¼ −gf ðA0E − A × B Þ: ð18Þ field quanta, such as photons or gluons, with momentum q ∈ ¯ 3 α ∈ 0 1 2 3 K× and polarization state f ; ; ; g.
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