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 constants 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. The space of momentum vectors is given by the simple cubic lattice Equation (13) can be used to eliminate Ba and Bc from B ¯ 3 q ∈ Z Eq. (16). After eliminating from the picture, we have the K× ¼f ¼ KLðz1;z2;z3Þjz1;z2;z3 g; ð24Þ evolution equations where KL is a lattice constant in momentum space, which is assumed to be small because it is given by the inverse ∂Aa a a abc b c ¼ −E þ ∇A0 − gf A A0; ð19Þ system size. The additional label a is associated with the ∂t infinitesimal generators of an underlying Lie group [assum- ing 3 values for SUð2Þ corresponding to the Wþ,W−, and and Z0 bosons mediating weak interactions, 8 values for SUð3Þ corresponding to the gluon “color octet” mediating strong ∂Ea interactions, and N2 − 1 values for general SUðNÞ]. For ¼ −Ja − ∇2Aa þ ∇∇ · Aa þ gfabcAbEc ∂t 0 simplicity, we occasionally refer to the gauge or vector bosons associated with the Yang-Mills field as gluons, for þ gfabc½2Ab · ∇Ac − Ab∇ · Ac þð∇AbÞ · Ac which a labels eight different “color” states. The gluon 2 − g fabsfcdsAb · AcAd; ð20Þ creation and annihilation operators satisfy canonical commutation relations. together with the relation (15). These equations define Details on the construction of Fock spaces can be found, classical Yang-Mills theories in a formulation that does not e.g., in Secs. 1 and 2 of [9], in Secs. 12.1 and 12.2 of [10],or manifestly exhibit Lorentz or gauge symmetry. These in Sec. 1.2.1 of [11]. We assume that the collection of the equations imply a conservation law with source term, states created by multiple application of all the above creation operators on a ground state, which is annihilated by all the corresponding annihilation operators, is complete. ∂ ˜ a0 J ˜ a abc b cμ þ ∇ · J ¼ −gf AμJ : ð21Þ In the following, we repeatedly need the properties of the ∂t polarization states. We hence give our explicit representations. The temporal unit four-vector, Aa 0 1 A wave equation for is obtained by subtracting 1 Eq. (16) from the time derivative of Eq. (14). Similarly, B C a0 0 a the wave equation for A is obtained by subtracting 0 B C ðn¯ μqÞ¼B C; ð25Þ Eq. (15) from the divergence of Eq. (14). We thus find the @ 0 A wave equations 0 q ∂2 is actually independent of . The three orthonormal spatial 2 a a abc bμ c c − ∇ A J gf A ∂0Aμ − 2∂μA polarization vectors are chosen as ∂ 2 0 ¼ 0 þ ð 0Þ t 0 1 2 abs cds b c d 0 þ g f f A · A A0 1 B C − δac∂ − abc b ∂ cμ 1 B q2 C ð 0 gf A0Þ μA ; ð22Þ ðn¯ μqÞ¼pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi B C; ð26Þ 2 2 @ − A q1 þ q2 q1 and 0

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TABLE I. Properties associated with the various creation pp. 239–240 of [3]. For quantum electrodynamics, all operators used for constructing the Fock space. the details have been elaborated in Sec. 3.2.5 of [11]. We here generalize these ideas to non-Abelian gauge Commuting Anticommuting theories. −metric þmetric −metric þmetric a0† aj† a† a† aq aq Bq Dq A. Ghost particle operators a† a† In the usual interpretation, the operators Bq and Dq in 0 1 Table I create massless ghost particles and their antipar- 0 ticles. The Fourier modes of the corresponding fields are B C hence given by 2 1 B q1q3 C n¯ μq pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi B C; 27 ð Þ¼ 2 2 @ A ð Þ q q1 þ q2 q2q3 1 1 a a† a a a† a − 2 − 2 cq ¼ pffiffiffiffiffiffi ðDq − B−qÞ; c¯q ¼ pffiffiffiffiffiffi ðBq þ D−qÞ: q1 q2 2q 2q and ð30Þ 0 1 0 If we further define the fields 1 B C 3 B q1 C rffiffiffi rffiffiffi ðn¯ μqÞ¼ B C; ð28Þ @ A q q q q2 _a a† a ¯_a a† − a cq ¼ i 2ðDq þ B−qÞ; cq ¼ i 2ðBq D−qÞ; q3 ð31Þ 1 2 where q ¼jqj. The polarization vectors n¯ q and n¯ q corre- 3 spond to transverse gluons, n¯ q corresponds to longitudinal the only nonvanishing anticommutation relations among gluons. Note the symmetry property the operators introduced in Eqs. (30) and (31) are given by ¯ α −1 α ¯ α n−q ¼ð Þ nq: ð29Þ a ¯_b − δ δ ¯a _b δ δ fcq; cq0 g¼ i ab qþq0;0; fcq; cq0 g¼i ab qþq0;0: In the BRST approach, one introduces the additional ð32Þ a† a a† a pairs Bq , Bq and Dq , Dq of creation and annihilation operators associated with ghost particles and their anti- Further note the adjointness properties with respect to the particles. They are assumed to satisfy canonical anticom- signed inner product, mutation relations. Fock spaces come with canonical inner products, scan. a‡ ¯a _a‡ ¯_a The superscript † on the creation operators can actually be cq ¼ c−q; cq ¼ c−q; ð33Þ interpreted as the adjoint with respect to the canonical inner product. In gauge theories, the canonical inner product is and the simple transformation rule for the energy of not the physical one. We need an additional signed inner noninteracting massless ghosts (neglecting an irrelevant product, ssign, for which states with negative norm exist (so constant to achieve normal ordering), that it is not an inner product in a strict sense). In both inner X X a† a a† a a _a 2 a a products, the natural base vectors of a Fock space, which qðBq Bq þ Dq DqÞ¼ ðc_qc¯−q þ q cqc¯−qÞ: ð34Þ are characterized by the occupation numbers of the various q∈ ¯ 3 q∈ ¯ 3 K× K× particles in the system, are orthogonal. If the canonical norm of a Fock base vector is one, the signed norm of that −1 At this point a comment on notation should be useful. vector is obtained by introducing a factor for every One might be tempted to interpret the dots in the symbols temporal gauge boson and for every ghost particle created _a ¯_a a† cq, cq introduced in Eq. (31) as time derivatives. However, by Bq . The adjoint of an operator with respect to the signed this interpretation works only for massless free ghost inner product is indicated by the superscript ‡. The particles. For quantum electrodynamics, this interpretation commutation and signed product properties of all particles would actually be justified. For non-Abelian gauge theo- are compiled in Table I. ries, however, the ghost particles are no longer free and a _a c_q, c¯q should be recognized as nothing but the operators IV. BASICS OF BRST APPROACH defined in Eq. (31); they do not coincide with time A brief discussion of BRST quantization in terms of derivatives that could be defined in terms of the full creation and annihilation operators can be found on Hamiltonian.

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B. Alternative representation of ghosts and c¯ by inverse factors and hence an additional conserved It has been pointed out by Kugo and Ojima [4] that the quantity in the ghost domain (see Sec. VI C for details). In lack of self-adjointness of the ghost operators expressed in view of Eqs. (33) and (39), we refer to the two options for “ ” Eq. (33) is a serious disadvantage in constructing the introducing ghost operators as cross-adjointness (option “ ” physical states. In non-Abelian Yang-Mills theories, it 1) and self-adjointness (option 2), respectively. We once moreover keeps the Hamiltonian from being self-adjoint. more emphasize that both options are realized on exactly Those authors hence recommend an alternative represen- the same Fock space. tation of ghosts that, however, requires noncanonical anticommutation relations for the creation and annihilation C. BRST charge and transformations a† a a† a operators Bq , Bq and Dq , Dq. A nonstandard construction To keep track of sums and differences of photon creation of a Fock space based on such noncanonical anticommu- and annihilation operators, to evaluate commutators and tation relations has been developed in [12]. Serious efforts anticommutators in an efficient manner, and to facilitate need to be made in order to introduce a proper vacuum state the comparison with the classical theory, we introduce the and a well-defined ghost number. operators In order to base our development on a well-defined 1 standard Fock space, we strongly prefer to keep canonical a 0 a0† a0 1 a1† a1 αq ¼ pffiffiffiffiffiffi ½n¯ qðaq − a−qÞþn¯ qðaq − a−qÞ anticommutation relations. Well-defined Fock spaces are 2q essential also for numerical calculations [13]. We therefore 2 a2† a2 3 a3† a3 þ n¯ qðaq þ a−qÞþn¯ qðaq − a−qÞ; ð41Þ propose a new representation of the ghost fields in terms of two creation and two annihilation operators, and rffiffiffi 1 a q 0 a0† a0 a3† a3 a a† a† a a εq ¼ −i ½n¯ qðaq þ a−q þ aq − a−qÞ cq ¼ pffiffiffi ðBq þ Dq − B−q þ D−qÞ; ð35Þ 2 2 q 1 a1† a1 2 a2† a2 þ n¯ qðaq þ a−qÞþn¯ qðaq − a−qÞ 1 a a† a† a a 3 a3† a3 a0† a0 c¯q ¼ pffiffiffi ðBq − Dq þ B−q þ D−qÞ; ð36Þ þ n¯ qðaq þ a−q þ aq − a−qÞ: ð42Þ 2 q a pffiffiffi According to Eqs. (3.73) and (3.81) of [11], αq corresponds i q a a a† a† a a to the four-vector potential and εq to the electric field c_q ¼ ðBq þ Dq þ B−q − D−qÞ; ð37Þ 2 [augmented by the time component introduced in Eq. (12)]. These four-vectors satisfy canonical commutation relations and pffiffiffi εaμ αbν ημνδ δ i q ½ p ; q ¼i ab pþq;0; ð43Þ ¯_a a† − a† − a − a cq ¼ ðBq Dq B−q D−qÞ: ð38Þ aμ bν aμ bν 2 ½αp ; αq ¼½εp ; εq ¼0; ð44Þ μν A straightforward calculation shows that the resulting where η represents the Minkowski metric, and possess the anticommutation relations for the operators defined in following self-adjointness properties with respect to the Eqs. (35)–(38) are identical to the previously found ones, signed inner product, where the only nontrivial anticommutators are given in a‡ a a‡ a αq ¼ α−q; εq ¼ ε−q: ð45Þ Eq. (32). Therefore, also most of the subsequent calculations for the two ways of introducing ghost fields are The energy of temporal and longitudinal noninteracting identical. Moreover, Eq. (34) remains valid. vector bosons can be written as The big advantage of the new definitions are the self- X a0† a0 a3† a3 adjointness properties qðaq aq þ aq aq Þ q∈ ¯ 3 K×X a‡ a a‡ a a‡ a _a‡ _a cq ¼ c−q; c_q ¼ c_−q; c¯q ¼ −c¯−q; c¯q ¼ −c¯−q: εajαa0 αajεa0 ¼ iqjð q −q þ q −qÞ q∈ ¯ 3 ð39Þ K× 1 X q q j k εajεak − εa0εa0 These properties imply the self-adjointness of products þ 2 2 q −q q −q : ð46Þ q∈ ¯ 3 q such as K× In the following, we use the notation a ¯b ‡ a ¯b ðcqcq0 Þ ¼ c−qc−q0 : ð40Þ 1 X ffiffiffiffi δ The systematic occurrence of cc¯ pairs is attractive also from ðABÞq ¼ p q0þq00;qAq0 Bq00 ; ð47Þ V q0 q00∈ ¯ 3 another viewpoint: We obtain a symmetry under rescaling c ; K×

074006-5 HANS CHRISTIAN ÖTTINGER PHYS. REV. D 97, 074006 (2018) for any two q dependent operators A and B. This notation BRST symmetry, is verified in Appendix A. As a somewhat for convolutions allows us to obtain more compact equa- simpler alternative, one can verify δQ ¼ 0 by means of the ¯ 3 tions, where working on the infinite lattice K× (that is, in BRST transformations (49). The first two lines of Eq. (49), the thermodynamic limit) is crucial for obtaining the usual which express the essence of the classical gauge trans- properties of convolutions (the relevance of high momenta formations (4), and the nilpotency of Q provide the deeper for a symmetry involving derivatives has been pointed reasons for writing the BRST charge in the form given out on p. 169 of [11]; on a finite lattice, rigorous BRST in Eq. (48). symmetry cannot be expected). The following expression for the BRST charge is V. CONSTRUCTION OF BRST inspired, for example, by Eq. (5.10) of [3] or Eq. (2.22) INVARIANT OPERATORS of [14], but the proper formulation in the present setting is not entirely straightforward, A simple way of constructing BRST invariant operators X is based on the identity a _a aj a Q ¼ ðε0qc−q − iqjεq c−qÞ q∈ ¯ 3 ½Q; ifQ; Xg ¼ 0; ð50Þ K× X 1 abc εaμαb − _a ¯b a ¯_b c which, for any operator X, follows trivially from the þ gf μ c c þ 2 c c c−q: ð48Þ q∈ ¯ 3 q nilpotency of Q. In other words, any operator ifQ; Xg is K× BRST invariant as it commutes with Q. In practice, Rearranging the factors in the three-particle collision one chooses X to produce a desirable term and one terms proportional to g in Eq. (48) is unproblematic automatically gets all the additional terms required for because, whenever a nonzero contribution might arise from BRST invariance. We illustrate the idea for some simple a nontrivial anticommutation relation, the corresponding choices of X, which consist of an α or ε paired with a c or c¯. _ structure constant with two equal labels vanishes. In To produce terms of the type c_ c¯, we choose particular, normal ordering is not an issue in three-particle X a _a collision terms, which is an appealing feature of such X1 ¼ α0qc¯−q: ð51Þ q∈ ¯ 3 interactions. K× The BRST charge (48) implies the following characteristic BRST transformations for non-Abelian gauge theories, Straightforward calculations based on the product rule and the results in Eq. (49) give a a abc b c δα0q ¼ c_q − gf ðα0c Þq; X a _a aj a0 ifQ; X1g¼ ðc_qc¯−q þ iq εq α−qÞ δαa − a − abc αb c j jq ¼ iqjcq gf ð j c Þq; q∈ ¯ 3 K× a abc b c X δεμq ¼ −gf ðεμc Þq: abc εajαb − _a ¯b αc þ gf ð j c c Þ−q 0q: ð52Þ 1 q∈ ¯ 3 δ a abc b c K× cq ¼ 2 gf ðc c Þq; a abc b c Similarly, to produce terms of the type cc¯, we choose δc_q ¼ gf ðc_ c Þq; X a a0 abc b c δc¯q ¼ εq þ gf ðc¯ c Þq; αa ¯a X2 ¼ iqj jqc−q; ð53Þ aj μ q∈ ¯ 3 δ¯_a − ε abc εb αc − _b ¯c b ¯_c K× cq ¼ iqj q þ gf ð μ c c þ c c Þq; ð49Þ and we obtain the BRST invariant operator where δ· ¼ i½Q; · for boson and δ· ¼ ifQ; ·g for fermion operators. The first two lines of Eq. (49) correspond to the X 2 a ¯a αajεa0 gauge transformation (4), the third line of Eq. (49) corre- ifQ; X2g¼ ðq cqc−q þ iqj q −qÞ q∈ ¯ 3 sponds to the transformation (11). K× X abc a ¯b c With the BRST charge (48) we have the tool to discuss all − gf iqjcqðc α Þ−q: ð54Þ q∈ ¯ 3 aspects of BRST symmetry. In particular, we can construct a K× BRST invariant Hamiltonian and the physical states in the large Fock space involving ghosts. The compatibility of Another interesting choice is given by BRST invariance with Lorentz symmetry is visible in the X a a first two lines of Eq. (49), which relates the BRST trans- X3 ¼ ε0qc¯−q: ð55Þ q∈ ¯ 3 formations of the four-vector potential to the time and space K× derivatives of the ghost field c. The nilpotency of the BRST charge (Q2 ¼ 0), which is crucial for the handling of It leads to the simple BRST invariant operator

074006-6 BRST QUANTIZATION OF YANG-MILLS THEORY: A … PHYS. REV. D 97, 074006 (2018) X a0 a0 ifQ; X3g¼− εq ε−q: ð56Þ The term proportional to g indeed cancels the contribution q∈ ¯ 3 from Φ in Eq. (59), however, a new second-order term in g K× arises. This second-order term has been written in a In summary, we have used bilinear operators X to compact form by arranging the three contributions resulting produce a number of BRST invariant operators ifQ; Xg. from the product rule in a standard form where, for A comparison with Eqs. (34) and (46) shows that, for rearranging the last contribution, the Jacobi identity is g ¼ 0, all our examples contain contributions from the free required. In this compact form we easily find the final Hamiltonian. This observation is very useful for the compensating term subsequent construction of a BRST invariant Hamiltonian. 2 abs cds X 00 g f f a b c d Φ δq q0 p p0 0α α 0 α α 0 ; 62 ¼ 4 þ þ þ ; jq kq jp kp ð Þ VI. YANG-MILLS HAMILTONIAN V q q0 p p0∈ ¯ 3 ; ; ; K× We now construct a BRST invariant Hamiltonian and discuss some implications. The construction is based in which all for spatial gauge boson operators are on an entirely on symmetry considerations. equal footing. In order to show that no leftover higher- order terms occur one needs to use the Jacobi identity. The Φ Φ0 Φ00 A. Construction of Hamiltonian operator þ þ hence is BRST invariant. The operator Φ contains the energy of the transverse We start from the energy of the noninteracting transverse gauge bosons and, according to Eq. (46), also part of polarizations of the vector bosons, energy of temporal and longitudinal gauge bosons. The X missing parts are found in the BRST invariant operators a1† a1 a2† a2 qðaq aq þ aq aq Þ established in Sec. V. Moreover, these BRST invariant q∈ ¯ 3 K× operators contain the energy (34) of noninteracting ghost X 1 1 1 2 2 2 aj aj particles. By collecting terms, we get the BRST invariant n¯ n¯ n¯ n¯ q αq αak εq εak : ¼ 2 ð jq kq þ jq kqÞð −q þ −qÞ ð57Þ total Hamiltonian q∈ ¯ 3 K× free coll A more convenient starting point is actually given by H ¼ Hgb þ Hgb ; ð63Þ

X 1 X q q Φ a1† a1 a2† a2 j k εajεak consisting of the energy of free massless gauge bosons and ¼ qðaq aq þ aq aq Þþ2 2 q −q q∈ ¯ 3 q∈ ¯ 3 q ghost particles K× K× 1 X X 2 aj ak aj aj aα† α a† a† ¼ ½ðq δ − q q Þαq α−q þ εq ε−q: ð58Þ free q a q a q a 2 jk j k Hgb ¼ qða aq þ B Bq þ D DqÞ; ð64Þ q∈ ¯ 3 q∈ ¯ 3 K× K×

A straightforward calculation yields and the interaction (confer to Eq. (3) of [15]), X Φ − abc 2δ − αaj αbk c i½Q; ¼ gf ðq jk qjqkÞ q ð c Þ−q; ð59Þ abc X ¯ 3 coll gf aj b c q∈K ffiffiffiffi δ 0 00 ε α α × Hgb ¼ p qþq þq ;0ð q jq0 0q00 V q;q0;q00∈K¯ 3 Φ ≠ 0 × so that is not yet BRST invariant for g .Asa a b c a b c a b c − c_qc¯ 0 α 00 − iq α α 0 α 00 − iq cqc¯ 0 α 00 Þ compensating term we consider q 0q j kq kq jq j q jq 2 abs cds X g f f a b c d abc X δq q0 p p0 0α α α α : gf þ 4 þ þ þ ; jq kq0 jp kp0 0 a b c V 3 Φ ffiffiffiffi δq q0 q00 0 −iq α α 0 α 00 ; 60 q q0 p p0∈ ¯ ¼ p þ þ ; ð jÞ kq kq jq ð Þ ; ; ; K× V q;q0;q00∈K¯ 3 × ð65Þ for which we find X By making sure that all four gauge boson polarizations 0 abc 2 aj bk c occur on an equal footing we have taken into account the i½Q; Φ ¼gf ðq δjk − qjqkÞαq ðα c Þ−q q∈ ¯ 3 Lorentz invariance of the free theory. The collisional K× 2 abs cds X contribution consists mostly of three-particle interactions, g f f δ but the second-order term in g represents a four-particle þ qþq0þpþp0;0 V q q0 p p0∈ ¯ 3 collision. Note that, for the construction of self-adjoint ; ; ; K× a b c d ghost particle operators, the Hamiltonian is self-adjoint ×iq cqα 0 α α 0 : 61 j kq jp kp ð Þ with respect to the signed inner product.

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a _a abc b c B. Evolution equations i½H;c¯q¼c¯q þ gf ðα0c¯ Þq: ð71Þ By recognizing i½H; A as the time derivative of an ¯_a operator A, we can now formulate the evolution equations This last equation clearly shows that the operator cq for various operators and, in particular, we can compare introduced in Eq. (31) is not the full time derivative of ¯a them to the classical equations compiled in Sec. II.We cq, as pointed out before. For non-Abelian gauge theories, begin with the evolution of the four-vector potential, interactions between ghost particles and vector bosons occur. More of these interactions are implied by a a a abc b c i½H; α q¼−ε q − iqjα0q − gf ðα α0Þq; ð66Þ a 2 a abc b c j j j i½H; c_q¼−q cq þ gf ðα0c_ Þq gfabc X which reassuringly is the quantum version of the Fourier ffiffiffiffi δ 00αbj c þ p q0þq00;qiqj q0 cq00 ; ð72Þ transformed classical evolution equation (19). For the V q0 q00∈ ¯ 3 ; K× temporal component of the four-vector potential, we find rffiffiffi and q αa 0† 0 −εa − αaj gfabc X i½H; 0q¼i ðaq þ a−qÞ¼ 0q iqj q ; ð67Þ ¯_a − 2 ¯a ffiffiffiffi δ αbj ¯c 2 i½H; cq¼ q cq þ p q0þq00;qiqj q0 cq00 : ð73Þ V q0 q00∈ ¯ 3 ; K× a which is the quantum version of the definition (12) of ε0q, relating εa to the time derivative of αa . 0q 0q C. Additional conserved charge We next turn to the electric field type operators and find It can easily be verified that an operator R possessing the εa 2αa − αak εa abc αbεc properties i½H; jq¼q jq qjqk q þiqj 0q þgf ð 0 j Þq X a a a a igfabc i½R; cq¼cq; i½R; c¯q¼−c¯q; ð74Þ ffiffiffiffi δ 2 0 00 αb − 0 αb αc þ p q0þq00;q½ð qk þqkÞ jq0 qj kq0 kq00 V q0 q00∈ ¯ 3 ; K× and 2 X g fabsfcds a a _a _a − δ αb αc αd i½R; c_q¼c_q; i½R; c¯q¼−c¯q; ð75Þ pþp0þp00;q kp kp0 jp00 V p p0 p00∈ ¯ 3 ; ; K× as well as vanishing commutators with all gauge boson gfabc X operators, commutes with the full Hamiltonian H. Such an − ffiffiffiffi δ 00 ¯b c p q0þq00;qiqj cq0 cq00 ; ð68Þ operator R can be interpreted as the generator of the V q0 q00∈ ¯ 3 ; K× symmetry associated with the rescaling of ghost operators with and without bars by reciprocal factors. where the same simplification as in Eq. (67) has been used. As suggested in Eq. (2.17) of [14], we have the following εa ¯b c Except for the terms involving 0 and c c , we recognize natural candidate for R, that exactly the same terms as in the classical evolution X a _a a a equation (20) in the absence of an external current. In view R ¼ ðcqc¯−q − c_qc¯−qÞ: ð76Þ a ε q∈ ¯ 3 of Eq. (12), 0 vanishes in the Lorenz gauge. Also the ghost K× term c¯bcc is clearly related to the proper handling of gauge conditions. The same remarks hold for the temporal One can easily verify by means of the anticommutation component relations (32) that this operator R indeed leads to the properties in Eqs. (74), (75). The simple form of R (compared aj to Eq. (2.17) of [14]) is a consequence of the fact that the dots H; εa q εq gfabc αbεcj c¯bc_c ; i½ 0q¼i j þ ð j þ Þq ð69Þ don’t indicate full time derivatives but rather the operators defined in Eq. (31) or Eqs. (37) and (38). which contains Eq. (15) in the absence of an external We further find the commutator current. The consistency of Eqs. (66)–(69) with the classical field equations implies that our minimal BRST i½R; Q¼Q: ð77Þ invariant extension of the Lorentz covariant free theory indeed leads to the quantized Yang-Mills field theory. On the kernel of Q, the operator R commutes also with the For completeness, we also look at the evolution equa- BRST charge. We can hence restrict the physical states to an tions for the ghost operators, eigenspace of R (most conveniently with eigenvalue zero).

a a VII. CONSTRUCTION OF PHYSICAL STATES i½H; cq¼c_q; ð70Þ In Sec. III, we had introduced two inner products, the and canonical and the signed ones, where the latter implies

074006-8 BRST QUANTIZATION OF YANG-MILLS THEORY: A … PHYS. REV. D 97, 074006 (2018) negative-norm states and hence is not truly an inner product. As the signed inner product serves for defining the physical bra-vectors in terms of ket-vectors, we need to identify a subspace of the full Fock space on which the signed product becomes a true inner product. If the ghost operators are introduced according to Eqs. (35)–(38), the BRST charge (48) is self-adjoint with respect to the signed inner product, Q‡ ¼ Q.However,the BRST charge is not self-adjoint with respect to the canonical inner product, Q† ≠ Q. We here decompose the Fock space into three mutually orthogonal spaces (in terms of the canonical inner product) and discuss the physical implications (associated with the signed inner product). FIG. 1. Decomposition of Fock space into three mutually orthogonal subspaces. A. Decomposition of states A decomposition of Fock space into three mutually where the three contributions to jφi are mutually orthogo- orthogonal subspaces is achieved in terms of the images nal in the canonical inner product. and kernels of the BRST operator Q and of the co-BRST operator Q†. Our discussion is based on Sec. 2.5 of [16]. B. BRST Laplacian A toy illustration of the essential features of the subsequent development is offered in Appendix B. Before we discuss the signed norm of states in the various Relying on the canonical inner product, the full Fock subspaces, we elaborate some details of the decomposition space F can be expressed as the direct sum of three (78). This can be done in terms of the BRST Laplacian mutually orthogonal spaces (the characteristic features of † 2 † † this decomposition are illustrated in Fig. 1), Δ ¼ðQ þ Q Þ ¼ QQ þ Q Q: ð82Þ Δ λ2 F ¼ðKerQ ∩ KerQ†Þ ⊕ ImQ ⊕ ImQ†: ð78Þ As is the square of a self-adjoint operator, its eigenvalues must be real and nonnegative. All states in KerQ ∩ KerQ† Δ In view of the nilpotency properties Q2 ¼ðQ†Þ2 ¼ 0,we are eigenstates of with eigenvalue zero. A nonzero have eigenvalue can only be obtained for vectors from one of the images, say jψi ∈ ImQ. To obtain Δjψi ≠ 0, jφi¼ † † † ψ ⊥ † ImQ ⊂ KerQ; ImQ ⊂ KerQ : ð79Þ Q j i has to lie in ðKerQÞ ¼ ImQ . Therefore, eigenstates of Δ with nonzero eigenvalues can only be obtained by As a next step, we prove the representations flipping back and forth between the two images when applying the two factors in the definition of Δ. By rescaling φ KerQ ¼ðImQ†Þ⊥; KerQ† ¼ðImQÞ⊥: ð80Þ j i, we find the following doublet of states, Q φ λ ψ We only prove the first identity in Eq. (80) because the j i¼ j i second one can be shown in exactly the same way. Q†jψi¼λjφi: ð83Þ If jφi∈KerQ, that is Qjφi¼0,wehave0¼scanðQjφi; jψiÞ¼scanðjφi;Q†jψiÞ for all jψi ∈ F. Conversely, if For λ ¼ 0 we would end up in the kernel of Δ,sothat jφi ∈ F is such that scanðjφi;Q†jψiÞ ¼ 0 for all † jψi ∈ F, then scanðQjφi; jψiÞ ¼ 0 for all jψi ∈ F, which KerΔ ¼ KerQ ∩ KerQ : ð84Þ implies Q φ 0. j i¼ † Equations (79) and (80) imply that ImQ and ImQ† are According to Eq. (83), both jφi ∈ ImQ and jψi ∈ ImQ 2 orthogonal spaces and that KerQ ∩ KerQ† is orthogonal to are eigenstates of Δ with the same eigenvalue λ . both images and exhausts the rest of F. We have thus We further note the property established Eq. (78) and the situation depicted in Fig. 1. † This equation implies that every state jφi ∈ F can be ðQ þ Q ÞðjφijψiÞ ¼ λðjψijφiÞ; ð85Þ written as so that jφijψi are eigenvectors of Q þ Q† with eigen- † † λ jφi¼jχiþQjψ 1iþQ jψ 2i with Qjχi¼Q jχi¼0; values . In summary, we have shown that the eigenvectors of Δ with nonzero eigenvalues are nicely organized in ð81Þ doublets.

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C. Physical subspace commuting with Q and Q†, the physical subspace KerΔ As a next step, we wish to identify KerΔ ¼ KerQ ∩ is invariant under Hamiltonian dynamics. KerQ† as the physical subspace of F in which ssign becomes a valid inner product, to be taken as the physical VIII. INCLUSION OF MATTER inner product. In other words, physical states are charac- In the terms proportional to g in the BRST charge (48) terized by Qjφi¼Q†jφi¼0. we recognize a term that contains the temporal component In the natural base of our Fock space, the canonical inner (17) of the current four-vector resulting from the gauge product is represented by the unit matrix, whereas the bosons. The simplest way to incorporate matter into the signed inner product is represented by a diagonal matrix σ BRST charge is to add the corresponding term involving with diagonal elements 1.IfA† and A‡ are the matrices the temporal component of the current four-vector asso- representing the two adjoints of an operator A in the natural † ‡ ciated with matter, basis, the definition of these adjoints implies σA ¼ A σ X σ2 1 σ ‡ †σ a0 a and, in view of ¼ , also A ¼ A . In view of the self- Q ¼ − Jq c−q: ð88Þ ‡ J q∈ ¯ 3 adjointness property Q ¼ Q, we conclude that we can K× choose a canonically orthonormal basis of eigenvectors of Q þ Q† which all possess signed norm þ1 or −1. This idea is consistent with the expression for the BRST We note that the signed norm of any state from one of charge in quantum electrodynamics (see, for example, the images of Q or Q† vanishes. For example, ssignðQjφi; Eq. (3.125) of [11]). Nilpotency of the extended BRST QjφiÞ ¼ ssignðjφi;Q‡QjφiÞ ¼ 0 as Q‡ ¼ Q and Q2 ¼ 0. charge Q þ QJ requires By superposition of states from the two images one can 2 Q þfQ; Q g¼0: ð89Þ produce states of negative norm. For the eigenvectors of J J † a0 Q þ Q found in Eq. (85),wehave Both terms can be calculated under the assumption that Jq commutes with all gauge boson and ghost operators: ssignðjφijψi; jφijψiÞ¼ðhφjψiþhψjφiÞ: ð86Þ X i abc c0 a b This result nicely shows that the (properly scaled) eigen- fQ; QJg¼2 gf Jq ðc c Þ−q; ð90Þ † q∈ ¯ 3 vectors of Q þ Q come in pairs with signed norms þ1 K× −1 and . and Can we now guarantee that the signed inner product is Δ ⊕ † ⊥ 1 X positive on Ker ¼ðImQ ImQ Þ ? The toy example of 2 a0 b0 a b Appendix B shows that the answer is “no.” In the BRST QJ ¼ 2 ½Jq ;Jq0 c−qc−q0 : ð91Þ q q0∈ ¯ 3 construction we have to make sure that the number of ; K× negative-norm states matches exactly the pairs in After using Eq. (47), the nilpotency condition can be † ImQ ⊕ ImQ . We need to verify that every negative-norm written as state can be written as a linear combination of zero-norm † gfabc states from ImQ and ImQ . a0 b0 − ffiffiffiffi c0 ½Jq ;Jq0 ¼ i p Jq q0 : ð92Þ The above construction is usually presented in two steps. V þ For the physical states, one first imposes the condition Qjφi¼0. By excluding ImQ† one avoids the above With this commutator we obtain the BRST construction of negative-norm states from the doublets transformation (83), but zero-norm states clearly still exist in ImQ.Ina a0 abc b0 c δJq ¼ −gf ðJ c Þq; ð93Þ second step, one considers equivalence classes of states that differ only by zero-norm states. Selecting representatives of which is exactly what we expect in view of the classical the equivalence classes is often referred to as gauge fixing. gauge transformation (7). To recover also the proper Our gauge fixing condition thus is Q†jφi¼0. According transformation behavior of the spatial components of the to Eq. (81), the states satisfying the first condition Qjφi¼ current four-vector, we need to generalize Eq. (92) to 0 possess the representation abc μ gf μ a0 b − ffiffiffiffi c † ½Jq ;Jq0 ¼ i p Jq q0 : ð94Þ jφi¼jχiþQjψi with Qjχi¼Q jχi¼0: ð87Þ V þ We can take jχi as the unique representative of an Equation (94) is a simple case of a current algebra, in equivalence class of states that differ by the zero-norm which neither an axial current nor a Schwinger term is states Qjψi for some jψi ∈ F, so that the physical sub- considered. Additional terms would require a change of the space indeed is KerΔ. The physical norms are independent BRST charge and/or Hamiltonian to make sure that the of the choice of the representative. For Hamiltonians BRST approach can still be used to handle the constraints

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free a0 − aj 0 associated with gauge theories. For example, the Schwinger i½Hmat;Jq iqjJq ¼ ð102Þ term [17] has been discussed in the context of Yang-Mills theory in Eq. (3.16) of [18] or Eq. (1.6) of [19].Itis is satisfied (see, e.g., Eq. (3.95) of [11] for quantum typically associated with sums that are not absolutely electrodynamics). In other words, the current four-vector convergent so that regularization is required. For electro- must be constructed such that a continuity equation holds. magnetic fields and massless fermions in one space However, the condition (102) is not a complete balance dimension (known as the Schwinger model [20]), the equation, which would require occurrence of the full free Schwinger term is finite and well defined. Possible mod- Hamiltonian instead of Hmat. An additional contribution ifications of the Hamiltonian and BRST charge are dis- a0 abc αb cμ cussed in Sec. 3.3 of [11]. As at least temporal photons i½HJ;Jq ¼gf ð μJ Þq ð103Þ have to acquire mass, the Schwinger term leads to chiral appears in the complete balance equation. In view of the symmetry breaking. classical continuity equation (21) one might assume that If we choose the Hamiltonian for the interaction of gauge this source term is compensated by including the current of bosons and matter as the field. However, if we define the quantum version of the X aμ a current four-vector, HJ ¼ − J−qαμq; ð95Þ q∈K¯ 3 ˜ a0 abc αbεcj × Jq ¼ gf ð j Þq; ð104Þ we find and εa − a i½HJ; μq¼ Jμq; ð96Þ ˜ aj − abc αbεcj cde αbαdαe Jq ¼ gf ½ð 0 Þq þ gf ð k j kÞq and hence the proper occurrence of the current in Eqs. (15) abc X igf 0 b 0 b c pffiffiffiffi δq0 q00 q q α 0 − q α 0 α 00 ; 105 and (16). In order to check whether H þ HJ is BRST þ þ ; ð j kq k jq Þ kq ð Þ V q0;q00∈K¯ 3 invariant with respect to the new BRST charge Q þ QJ,we × calculate the commutators a lengthy calculation shows that additional gauge terms X a _a − a a appear, i½Q; HJ¼ ðJ0qc−q iqjJjqc−qÞ q∈ ¯ 3 0 aj K× ˜ a − ˜ X i½H þ HJ; Jq iqjJq abc bμ c a − gf J c αμq; 97 abc b cμ ð Þ−q ð Þ ¼ −gf ðαμJ Þq q∈ ¯ 3 K× gfabc X X ffiffiffiffi b c0 0 b cj þ p δq0 q00 qfi½H; ε 0 α 00 − iq ε 0 α 00 − a _a þ ; 0q q j 0q q i½QJ;H¼ J0qc−q; ð98Þ V q0 q00∈ ¯ 3 ; K× q∈K¯ 3 × − bde _d ¯eαc0 − 0 d ¯eαcj gf ½cq0 ðc Þq00 iqjcq0 ðc Þq00 g: ð106Þ and X Therefore, a more careful analysis of say color conservation abc bμ c a on the physical space is required. i½QJ;HJ¼gf ðJ c Þ−qαμq: ð99Þ q∈ ¯ 3 K× IX. SUMMARY AND DISCUSSION The incomplete compensation of terms in Eqs. (97)–(99) We have elaborated all the details of the BRST quan- implies that H þ HJ is not BRST invariant with respect to tization of Yang-Mills theory in a strictly Hamiltonian the new BRST charge Q þ QJ. This is not surprising as the Hamiltonian for noninteracting matter is still missing. approach. A new representation of ghost-field operators in BRST invariance of the full Hamiltonian requires terms of canonical creation and annihilation operators has X been introduced in Eqs. (35)–(38). This representation free a a combines two pivotal advantages: (i) there exists a well- i½QJ;Hmat¼ iqjJjqc−q; ð100Þ q∈ ¯ 3 defined Fock space that serves as the underlying Hilbert K× space for carrying out the Hamiltonian approach and (ii) the or, by means of Eq. (88), BRST charge (48) and the Hamiltonian (63)–(65) turn out X to be self-adjoint operators with respect to the physical free a0 − aj a 0 ði½Hmat;Jq iqjJq Þc−q ¼ : ð101Þ inner product. The Fock space actually carries two inner q∈ ¯ 3 K× products, a canonical and a signed one, where the restric- tion of the latter to a suitable subspace serves as the The current four-vector must be defined such that the physical inner product. The occurrence of negative-norm operator identity states in the unrestricted space explains why ghost particles

074006-11 HANS CHRISTIAN ÖTTINGER PHYS. REV. D 97, 074006 (2018) X εa _a − εaj a can combine the anticommutation relations for fermions Q0 ¼ ð 0qc−q iqj q c−qÞ; ðA1Þ with zero spin, in an apparent violation of the spin-statistics q∈ ¯ 3 K× theorem. X The construction of the BRST charge is based on the abc aμ b c Q1 ¼ gf ðε αμÞqc−q; ðA2Þ following two properties: (i) its role as the generator of the q∈ ¯ 3 operator version of classical gauge transformations and K× (ii) its nilpotency. Lorentz symmetry is taken into account X abc a b c in the construction of the Fock space (with four gauge Q2 ¼ −gf ðc_ c¯ Þqc−q; ðA3Þ q∈ ¯ 3 boson polarizations), in the formulation of the BRST K× charge (inherited from classical gauge transformations), and in the construction of the Hamiltonian as the minimal and BRST invariant extension of the free theory (in which all 1 X four gauge boson polarizations are on an equal footing). abc _a b c Q3 ¼ gf c¯−qðc c Þ : ðA4Þ The final Hamiltonian found in Eqs. (63)–(65) reproduces 2 q q∈K¯ 3 the field equations of classical Yang-Mills theory. × The interaction with matter has been included in terms of Based on trivial canonical commutation and anticommu- the current four-vector by adding the contributions (88) and tation relations, we find (95) to the BRST charge and Hamiltonian, respectively. In order to make the BRST approach work, a current algebra 2 0 has to be postulated, where a particularly simple one is Q0 ¼ ; ðA5Þ given in Eq. (94). Any change in the current algebra, say by 2 a Schwinger term, requires modifications of the BRST exactly as for Abelian gauge theory. The evaluation of Q1 charge and the BRST invariant Hamiltonian. We have also is based on the commutation relations (43) and (44); the discussed the formulation of the conservation law associ- result is ated with BRST symmetry. X 2 2 ads bcs dμ c b a Whereas the Hamiltonian approach to BRST quantiza- Q1 ¼ −ig f f ðε αμÞ−qðc c Þq: ðA6Þ q∈ ¯ 3 tion on Fock space has the educational advantage of K× being nicely explicit and transparent, it has serious dis- advantages in practical calculations. In particular, pertur- By means of Eq. (32) we find bation theory becomes very cumbersome (see Appendix A X of [11] for some simplifications). Our motivation for 2 2 cds abs a d b c Q2 ¼ ig f f ðc¯ c_ Þqðc c Þ−q: ðA7Þ elaborating the Hamiltonian approach stems from the q∈ ¯ 3 K× formulation of dissipative quantum field theory [11,21], which is based on quantum master equations for evolving To obtain density matrices in time as an irreversible generalization of Hamiltonian dynamics. Dissipative quantum field 2 Q ¼ 0; ðA8Þ theory, which is based on the idea that the elimination 3 of degrees of freedom in renormalization procedures leads we need the Jacobi identity (3). to the emergence of irreversibility, adds rigor, robustness As a consequence of and intuition to the field and hence has the potential to clarify the foundations of quantum field theory. Dissipation 0 can easily be introduced in the Hamiltonian approach (see fQ1;Q2g¼ ; ðA9Þ

Sec. 1.2.3.2 of [11]). 2 Stochastic simulation techniques developed for quan- the nonzero contribution to Q1 in Eq. (A6) can only be tum master equations [22] can then be used to simulate compensated by quantum field theories (see Secs. 1.2.8.6 and 3.4.3.3 of i X [11] for details). With the present paper, the simulation − 2 cds abs εdμαc b a fQ1;Q3g¼ 2 g f f ð μÞ−qðc c Þq: ðA10Þ ideas so far applied only in a rudimentary way to q∈K¯ 3 quantum electrodynamics [23], become applicable to × quantum chromodynamics. Indeed, the Jacobi identity (3) implies

APPENDIX A: PROOF OF Q2 =0 2 Q1 þfQ1;Q3g¼0: ðA11Þ In order to prove the nilpotency of the BRST charge 2 defined in Eq. (48), we write Q ¼ Q0 þ Q1 þ Q2 þ Q3 The nonzero contribution to Q2 in Eq. (A7) can only be with compensated by

074006-12 BRST QUANTIZATION OF YANG-MILLS THEORY: A … PHYS. REV. D 97, 074006 (2018) 8 0 1 9 i X 1 2 ads bcs ¯a _d b c < = fQ2;Q3g¼2 g f f ðc c Þqðc c Þ−q: ðA12Þ B C q∈ ¯ 3 ImQ ¼ λ@ 1 A λ ∈ R ; ðB2Þ K× : ; 0 By once more using the Jacobi identity (3), we find

2 and Q þfQ2;Q3g¼0: ðA13Þ 2 8 0 1 0 1 9 1 0 < = We still need to evaluate the anticommutators of Q0 with B C B C KerQ ¼ λ@ 1 A þ λ0@ 0 A λ; λ0 ∈ R ; ðB3Þ all the other contributions to Q. We first evaluate : ; X 0 1 abc a b c fQ0;Q1g¼igf ε0qðc_ c Þ−q q∈ ¯ 3 illustrating the general relation Im Q ⊂ Ker Q. We similarly K× X have 1 aj þ gfabc q εq ðcbccÞ : ðA14Þ 8 0 1 9 2 j −q 1 q∈ ¯ 3 < = K× † B C ImQ ¼ λ@ −1 A λ ∈ R ; ðB4Þ : ; The next anticommutator is given by 0 X 8 0 1 0 1 9 abc a b c 1 0 fQ0;Q2g¼−igf ε ðc_ c Þ−q; ðA15Þ < = 0q B C B C q∈ ¯ 3 † 0 0 K× KerQ ¼ λ@ −1 A þ λ @ 0 A λ; λ ∈ R ; ðB5Þ : ; 0 1 which cancels the first term on the right-hand side of Eq. (A14). A final straightforward calculation yields and ImQ† ⊂ KerQ†. We also find ImQ† ¼ðKerQÞ⊥ and † ⊥ 1 X ImQ ¼ðKerQ Þ , implying the orthogonality of ImQ and − abc εaj b c † Δ † 2 † † fQ0;Q3g¼ 2 gf qj q ðc c Þ−q: ðA16Þ ImQ . The kernel of ¼ðQ þ Q Þ ¼ QQ þ Q Q, q∈K¯ 3 0 1 × 400 B C As fQ0;Q3g cancels the second term on the right-hand side Δ ¼ @ 040A; ðB6Þ of Eq. (A14), we obtain 000 0 fQ0;Q1gþfQ0;Q2gþfQ0;Q3g¼ : ðA17Þ coincides with KerQ ∩ KerQ†, so that KerΔ,ImQ and ImQ† are three mutually orthogonal spaces. The total space By summing up Eqs. (A11), (A13), (A17) and using the is the direct sum of these three vector spaces. Any vector results (A5), (A8), (A9), we indeed obtain the desired 2 0 can uniquely be written as the sum of three vectors, one nilpotency of the BRST charge defined in Eq. (48), Q ¼ . from each of these spaces. Introducing a signed inner product by 0 1 APPENDIX B: CONSTRUCTION OF PHYSICAL −100 STATES—A TOY VERSION B C σ ¼ @ 010A; ðB7Þ We here sketch a toy version of the construction of 001 physical states in the BRST approach. The general idea is nicely illustrated in a three-dimensional cartoon version of we realize the self-adjointness property Q‡ ¼ σQ†σ ¼ Q. temporal, longitudinal and transverse gauge bosons (where For this signed inner product, states from ImQ or ImQ† physical states do not contain any right bosons, equivalent have zero norm, states from KerΔ have positive norm. Had physical states differ by left bosons, and unique represent- we chosen a signed inner product with atives of equivalence classes do not contain any left photons). 0 1 For the linear operators −10 0 B C 0 1 0 1 σ0 @ 01 0A 1 −10 110 ¼ ; ðB8Þ B C B C 00−1 Q ¼ @ 1 −10A;Q† ¼ @ −1 −10A; ðB1Þ 000 000 we would still have the self-adjointness property Q‡ ¼ σ0 Q†σ0 ¼ Q. However, we would have introduced too many one easily verifies Q2 ¼ðQ†Þ2 ¼ 0. The image and kernel states with a negative norm leading to an unphysical inner of the operator Q are given by product on KerΔ.

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