Exactly solvable models and spontaneous symmetry breaking
aa r X i v : . [ h e p - t h ] N ov Few Body Systems manuscript No. (will be inserted by the editor)
L’ubom´ır Martinovi˘c
Exactly solvable models and spontaneous symmetrybreaking ⋆ Abstract
We study a few two-dimensional models with massless and massive fermions in the hamil-tonian framework and in both conventional and light-front forms of field theory. The new ingredientis a modification of the canonical procedure by taking into account solutions of the operator fieldequations. After summarizing the main results for the derivative-coupling and the Thirring models,we briefly compare conventional and light-front versions of the Federbush model including the massivecurrent bosonization and a Bogoliubov transformation to diagonalize the Hamiltonian. Then we sketchan extension of our hamiltonian approach to the two-dimensional Nambu – Jona-Lasinio model andthe Thirring-Wess model. Finally, we discuss the Schwinger model in a covariant gauge. In particular,we point out that the solution due to Lowenstein and Swieca implies the physical vacuum in terms of acoherent state of massive scalar field and suggest a new formulation of the model’s vacuum degeneracy.
Keywords solvable relativistic models · operator solutions · vacuum structure · Bogoliubovtransformation · spontaneous symmetry breaking Exactly solvable models may seem to be almost a closed chapter in the development of quantum fieldtheory. Our original aim for looking at this class of simple relativistic theories was to learn about theirproperties in the light-front (LF) formulation and compare that picture with the standard one (bythe latter we mean conventional operator formalism in terms of usual space-like (SL) field variables).Surprisingly enough, a closer look at these models revealed certain inconsistencies and contradictionsin the known SL solutions. For example, the Fock vacuum was taken as the true ground state forcalculations of the correlation functions in the Thirring model [1] although a derivation of the model’sHamiltonian shows that it is non-diagonal when expressed in terms of corresponding creation andannihilation operators. Hence the Fock vacuum cannot be its (lowest-energy) eigenstate. Anotherexample is the (massless) Schwinger model, often invoked as a prototype for more complicated gaugetheories. It turns out that the widely-accepted covariant-gauge solution [2] involves some unphysicaldegrees of freedom as a consequence of residual gauge freedom. A rigorous analysis [3] in the axiomaticspirit showing some spurious features of the solution [2] remained almost unnoticed and a less formal,e.g. a hamiltonian study correcting the physical picture seems to be lacking in literature.Defining property of the soluble models is that one can write down operator solutions of the fieldequations. Hence, one should be able to extract their physical content completely. A novel feature that ⋆ Presented by the author at LIGHTCONE 2011, 23–27 May, 2011, DallasInstitute of Physics, Slovak Academy of Sciences, D´ubravsk´a cesta 9, 845 11 Bratislava, SlovakiaTel.:+421-2-5941 055, Fax: +421-2-5477 6085E-mail: [email protected]
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BLTP JINR, 141 980 Dubna, Russia has been overlooked so far is a necessity to formulate these models in terms of true field degrees offreedom – the free fields. As we show below, the operator solutions are always composed from free fields.One should take this information into account by re-expressing the Lagrangiam and consequently theHamiltonian in terms of these variables. One can summarize the above statement by a slogan: ”workwith the right Hamiltonian!” Recently, we applied this strategy to the massive derivative-couplingmodel (DCM), the Thirring and the Federbush models. The brief summary of the achieved results isgiven in the following section. Then we extend our approach to the case of the chiral Gross-Neveumodel, the Thirring-Wess and the Schwinger model. Not all the details are worked out at the presentstage – we formulate the main ideas and indicate the strategy to be followed. The full treatment ofthe models will be given separately [4]. We conclude the present paper with a brief description of thesymmetry-breaking pattern fully based on the light-front dynamics. [5] turns out to be almost a trivial one. Its Lagrangian L = i Ψ γ µ ↔ ∂ µ Ψ − mΨ Ψ + 12 ∂ µ φ∂ µ φ − µ φ − g∂ µ φJ µ , J µ = Ψ γ µ Ψ. (1)leads to the Dirac equation which is explicitly solved in terms of a free scalar and fermion field: Ψ ( x ) =: e igφ ( x ) : ψ ( x ) , iγ µ ∂ µ ψ ( x ) = mψ ( x ) . (2)The massive scalar field φ ( x ) obeys the free Klein-Gordon equation as a consequence of the currentconservation. The conventional canonical treatment yields a surprising result: the obtained LF Hamil-tonian is a free one while the SL Hamiltonian contains an interacting piece, which is non-diagonalin terms of Fock operators and hence its true ground state (which can be obtained by a Bogoliubovtransformation) differs from the Fock vacuum. So the physical pictures in two quantization schemescontradict each other. The explanation is simple. One observes that the solution (2) means that thereis no independent interacting field - it is composed from the free fields. We have to insert the solution tothe Lagrangian first (analogously to inserting a constraint into a Lagrangian), then calculate conjugatemomenta and derive the Hamiltonian. In this way, a free Lagrangian and Hamiltonian are found alsoin the SL case. The new procedure does not alter the LF result. The correlation functions in the twoschemes coincide as well. They are built from free scalar and fermion two-point functions. The Thirring model [6] with its Lagrangian describing a self-interacting massless Fermi field L = i Ψ γ µ ↔ ∂ µ Ψ − gJ µ J µ , J µ = Ψ γ µ Ψ (3)is a more complicated theory. The simplest solution is similar to Eq.(2) but the elementary scalar field φ is replaced by the composite field j ( x ) defined via J µ = j µ = √ π ∂ µ j . The corresponding Hamiltonian H is non-diagonal in composite boson operators c, c † built from fermion bilinears according to j µ ( x ) = − i √ π Z dk √ k k µ (cid:8) c ( k ) e − i ˆ k.x − c † ( k ) e i ˆ k.x (cid:9) , (4) c ( k ) = i √ k Z dp (cid:8) θ (cid:0) p k (cid:1)(cid:2) b † ( p ) b ( p + k ) − ( b → d ) (cid:3) + ǫ ( p ) θ (cid:0) p ( p − k ) (cid:1) d ( k − p ) b ( p ) (cid:9) . (5)A diagonalization by a Bogoliubov transformation U HU − implemented by a unitary operator U [ γ ( g )],where γ is a suitably chosen function, generates the true ground state as | Ω i = N exp (cid:8) − κ + ∞ Z −∞ d p c † ( p ) c † ( − p ) (cid:9) | i . (6)Here N is a normalization factor and κ is a g -dependent function. | Ω i corresponds to a coherent stateof pairs of composite bosons with zero values of the total momentum, charge and axial charge. Thusno chiral symmetry breaking occurs in the model at least for g ≤ π where the diagonalization is valid. The Federbush model [7] is the only known massive solvable model. The solvability comes from aspecific current-current coupling between two species of massive fermions described by the Lagrangian L = i Ψ γ µ ↔ ∂ µ Ψ − mΨΨ + i Φγ µ ↔ ∂ µ Φ − µΦΦ − gǫ µν J µ H ν . (7)Here the currents are J µ = Ψ γ µ Ψ, H µ = Φγ µ Φ . The coupled field equations iγ µ ∂ µ Ψ ( x ) = mΨ ( x ) + gǫ µν γ µ H ν ( x ) Ψ ( x ) , iγ µ ∂ µ Φ ( x ) = µΦ ( x ) − gǫ µν γ µ J ν ( x ) Φ ( x ) (8)have the solution of the form (2) with two ”integrated currents” j ( x ) and h ( x ). One again finds thatthe structure of the SL and LF Hamiltonians coincides only when the operator solution is imple-mented in the Lagrangians. However, the SL Hamiltonian is not diagonal and a Bogoliubov transfor-mation is needed to find the physical ground state. This requires a generalization of Klaiber’s masslessbosonization yielding a complicated substitute for c ( k ) of Eq.(5). In a sharp contrast, the LF massivebosonization is as simple as the SL massless one. The model is very suitable for a non-perturbativecomparison of the two forms of the relativistic dynamics. This is because 2-D massless fields cannot betreated directly in the LF formalism (only as the massless limits of massive theories - this is obviousalready from the LF massive two-point functions). Exponentials of the massive composite fields aremore singular than the massless once. They have to be defined using the ”triple-dot ordering” [8,9]which generalizes the normal ordering (subtractions of the VEVs order by order). We avoid this bybosonization of the massive current. The price we pay is complicated commutators at unequal timesthat are needed for computation of correlation functions. In any case, the correlators in the SL and LFversion of the theory should coincide in form. A remarkable albeit for the moment only a conjecturedscenario is that this will indeed happen with complicated operator structures plus non-trivial vacuumstructure in the SL case and with much simpler operator part plus the Fock vacuum in the LF case. The Lagrangian and field equations of the chiral Gross-Neveu model [10] are L = i Ψ γ µ ↔ ∂ µ Ψ − g (cid:2)(cid:0) Ψ Ψ (cid:1) + (cid:0) Ψ iγ Ψ (cid:1) (cid:3) , iγ µ ∂ µ Ψ = g (cid:2)(cid:0) Ψ Ψ (cid:1) Ψ − (cid:0) Ψ γ Ψ (cid:1) γ Ψ (cid:3) . (9)This theory is a 2-D version of the Nambu–Jona-Lasinio model. When rewritten in the componentform, one realizes that the above equations up to the sign coincide with those in the Thirring model: i (cid:16) ∂ − ∂ (cid:17) Ψ = 2 gΨ † Ψ Ψ = − gΨ † Ψ Ψ = − g ( j + j ) ,i (cid:16) ∂ + ∂ (cid:17) Ψ = 2 gΨ † Ψ Ψ = − gΨ † Ψ Ψ = − g ( j − j ) . (10)Inserting the solution known from the Thirring model into the Lagrangian, we find the Hamiltonian H = + ∞ Z −∞ d x n − iψ † α ∂ ψ + 12 g (cid:0) j j − j j (cid:1) + g h(cid:16) ψψ (cid:17) − (cid:16) ψγ ψ (cid:17) i . (11)We can bosonize the scalar densities Σ ( x ) = ψψ = ψ † ψ + ψ † ψ , Σ ( x ) = iψγ ψ = i (cid:0) ψ † ψ − ψ † ψ (cid:1) : Σ ( x ) = + ∞ Z −∞ dk π h A ( k , t ) e ik x + A † ( k , t ) e − ik x i , Σ ( x ) = + ∞ Z −∞ dk π h B ( k , t ) e ik x + H.c. i (12)With the Fock expansions for the Fermi field and after the Fourier transform, we obtain A ( k , t ) = + ∞ Z −∞ d p n h b † ( − p ) b ( k − p ) + d † ( − p ) d ( k − p ) i θ (cid:0) p ( k − p ) (cid:1) e i p t − θ ( p k ) ǫ ( p ) (cid:2) d ( − p ) b ( p + k ) + b ( − p ) d ( p + k ) (cid:3) e − i p t o e − ik t , (13) and similarly for B ( k , t ). One then has to diagonalize the Hamiltonian H = H + H + H where H = − gπ + ∞ Z −∞ dk | k | h c † ( k ) c † ( − k ) + c ( k ) c ( − k ) i ,. H = − g π + ∞ Z −∞ dk | k | nh A † ( k ) A ( k ) + A † ( k ) A † ( − k ) + A ( k ) A ( − k ) i + h A → B io . (14)Obviously | i is not an eigenstate of H . The true vacuum has to be found by a Bogoliubov transforma-tion. It will be different than the Thirring-model vacuum and probably non-invariant under Q . Thisremains to be verified. We also leave for future work generalization of the model to N f flavours. This model [11] is simpler than the Schwinger model because the nonzero bare mass of the vector fieldremoves gauge invariance with all its subtleties. The corresponding Lagrangian L = i Ψ γ µ ↔ ∂ µ Ψ − G µν G µν + µ B µ B µ − eJ µ B µ , G µν = ∂ µ B ν − ∂ ν B µ . (15)leads to the field equations – the Dirac and Proca equations: iγ µ ∂ µ Ψ ( x ) = eγ µ B µ ( x ) Ψ ( x ) , ∂ µ G µν ( x ) + µ B µ ( x ) = eJ µ ( x ) . (16)The rhs of the second equation reduces to ( ∂ ν ∂ ν + µ ) B ν since ∂ ν B ν = 0 due to the current conservation.The latter condition also permits us to write down the solution of the corresponding Dirac equation as Ψ ( x ) = exp n − ie γ ∞ Z −∞ d y ǫ ( x − y ) B ( y , t ) o ψ ( x ) , γ µ ∂ µ ψ ( x ) = 0 . (17)Product of two fermion operators has to be regularized by a point-splitting. The integral in the exponentcontributes naturally to find ( j µ and j µ are the free currents) J µ ( x ) = j µ ( x ) − eπ B µ ( x ) , J µ ( x ) = j µ ( x ) − eπ ǫ µν B ν ( x ) . (18)Inserting the above J µ ( x ) to the Proca equation, one finds that the the bare mass is replaced by µ = µ + e /π and that this equation can be easily inverted since only the free fields are involved.Thus, there is no dynamically independent vector field. Following our method, we insert the solutionsfor B µ and Ψ into Lagrangian and then derive the Hamiltonian. The question if the latter will bediagonal or will have to be diagonalized, together with other properties of the model, is under study. The masslessness of the vector field makes the Schwinger model more subtle than was the previousone. The key question is to correctly handle the gauge variables since in the covariant gauge ∂ µ A µ = 0not all gauge freedom has been removed. We implement the gauge condition in the Lagrangian as [12]: L = i Ψ γ µ ↔ ∂ µ Ψ − F µν F µν − eJ µ ( x ) A µ ( x ) − G ( x ) ∂ µ A µ ( x )+ 12 (1 − γ ) G ( x ) , F µν = ∂ µ A ν − ∂ ν A µ . (19)The gauge-fixing terms furnish the component A with the conjugate momentum, Π A ( x ) = − G ( x ).Moreover, they guarantee restriction to an arbitrary covariant gauge in which neither the condition ∂ µ A µ ( x ) = 0 nor the Maxwell equations ∂ µ F µν ( x ) = eJ ν ( x ) hold as operator relations. The gaugefixing field obeys ∂ µ ∂ µ G ( x ) = 0 so that positive and negative-frequency parts G ( ± ) ( x ) are well defined. Our strategy is to proceed in the spirit of K. Haller’s generalization [12] of the Gupta-Bleulerquantization, in which the unphysical components of the gauge field are represented as ghost degreesof freedom of zero norm, carrying vanishing momentum and energy. To ensure that we are dealing withthe original 2-dimensional QED, we have to restrict the theory to the physical subspace G (+) | phys i = 0.We will choose γ = 1 in the above Lagrangian. Then the gauge condition is an operator relationwhile the (modified) Maxwell and Dirac equations read ∂ µ F µν ( x ) = eJ ν ( x ) − ∂ ν G ( x ) , iγ µ ∂ µ Ψ ( x ) = eγ µ A µ ( x ) Ψ ( x ) . (20)The solution of the latter is completely analogous to the Thirring-Wess model case, Eq.(17). Again, thevector and axial-vector currents have to be calculated via point-splitting with an important difference:the exponential of the line integral of the gauge field must be inserted in the current definition tocompensate for the violation of gauge invariance due to the point splitting. After inserting the calculatedinteracting current into the Maxwell equations, we have to express the Lagrangian and Hamiltonianin terms of the free fields as before. The physical picture will become transparent if we make a unitarytransformation to the Coulomb-gauge representation [12]. Before performing this step, let us indicatethe main ingredients of the original covariant-gauge solution [2,13] and point out a problem with it.The starting point was the Ansatz for the gauge field and the currents, ( ˜ ∂ µ = ǫ µν ∂ ν ) A µ = − √ πe (cid:0) ˜ ∂ µ Σ + ∂ µ ˜ η (cid:1) , J µ = Ψ γ µ Ψ = − √ π ˜ ∂ µ Φ, J µ = Ψ γ µ γ Ψ = − √ π ∂ µ Φ, (21)where Σ, ˜ η and Φ are so far unspecified scalar fields. In the ∂ µ A µ = 0 gauge, one finds ∂ µ ∂ µ ˜ η = 0 and F µν = √ πe ǫ µν ∂ ρ ∂ ρ Σ. From the anomalous divergence of the axial current ∂ µ J µ = e π e µν F µν (22)one concludes that ∂ µ ∂ µ Φ = ∂ µ ∂ µ Σ or Φ = Σ + h with the free massless field h obeying ∂ µ ∂ µ h = 0.Then the vector current is J µ = − √ π ˜ ∂ µ + L µ , L µ = − √ π ˜ ∂ µ h. (23)From the Maxwell eqs. ˜ ∂ ν (cid:0) ∂ ρ ∂ ρ + e π Σ (cid:1) − e √ π L ν = 0 (24)one concludes that ˜ ∂ µ L µ = 0 or (cid:0) ∂ ρ ∂ ρ + e π (cid:1) Σ = 0 . (25)One component of the gauge field, namely Σ ( x ), became massive. For consistency reasons, L µ canvanish only weakly, h ψ | L µ ( x ) | ψ i = 0. With the above Ansatz for A µ , Dirac eq. becomes iγ µ ∂ µ Ψ = −√ πγ µ γ ∂ µ (cid:0) Σ + η (cid:1) Ψ with the solution Ψ ( x ) =: e i √ πγ (cid:0) Σ ( x )+ η ( x ) (cid:1) : ψ ( x ) . (26)Calculation of the currents via the point-splitting yields identification h ( x ) = η ( x ) + ϕ ( x ), where ϕ ( x )is the ”potential” (integrated current) of the free currents. From [ A ( x ) , π ( y )] = iδ ( x − y ) andwith π = F = e √ π Σ one gets the equal-time commutator [ Σ ( x ) , ∂ Σ ( y )] = iδ ( x − y ) of acanonical scalar field. Inserting Eq.(26) to the original Lagrangian, we derive the physical part of theHamiltonian as : H = + ∞ Z −∞ d x h − iψ † α ∂ ψ + 12 e π Σ i . (27)This Hamiltonian is non-diagonal when expressed in terms of Fock operators of Σ ( x ). A Bogoliubovtransformation is necessary. A coherent-type of vacuum state will be obtained as the true vacuum.An interesting aspect of the model is its vacuum degeneracy and the theta vacuum. In the work[2] the mechanism generating multiple vacua is based on ”spurion” operators. These however have been shown to be an artifact of the incorrect treatment of residual gauge freedom [3]. What canbe the true mechanism of the vacuum degeneracy in the Scwinger model? We believe that it is thepresence of a gauge zero mode in the finite-volume treatment of the model [14] together with a quantumimplementation of residual invariance under large gauge transformations as desribed in [15]. This leadsus naturally to the finite-volume reformulation of our approach outlined in the first part of this section. At the LC workshop in Valencia, Marvin Weinstein criticised the way how the LF theory describesspontaneous symmetry breaking, saying that Goldstone (or hidden) symmetry is a chalenge for LFtheory. Where is vacuum degeneracy? Actually the latter can be described if one takes into account amechanism based on the presence of dynamical fermion zero modes [16]. O (2)-symmteric sigma modelprovides us with a good example. Its Hamiltonian is symmetric under axial-vector transformations Ψ + ( x ) → e − iβγ Ψ + ( x ) = V ( β ) Ψ + ( x ) V − ( β ) , V ( β ) = exp {− iβQ } , Q = Z V d xJ +5 ( x ) . (28)The operator of the axial charge will not annihilate the LF vacuum since in addition to the normal-mode part (which annihilates it) it contains also the zero-mode term, Q = Q N + Q , where Q = X p ⊥ ,s s h(cid:16) b † ( p ⊥ , s ) d † ( − p ⊥ , − s ) + H.c. (cid:17) + b † ( p ⊥ , s ) b ( p ⊥ , s ) − d † ( p ⊥ , s ) d ( p ⊥ , s ) (cid:3) . (29)The term b † ( p ⊥ ) d † ( − p ⊥ ) will generate an infinite set of degenerate vacuum states. One has all prop-erties for deriving the Goldstone theorem in the usual way. We conclude with the statement that the realm of exactly solvable models still offers us certainsurprises and room for improvement. And that degenerate vacua exist in the LF formalism in spite of itskinematically defined vacuum state.
Acknowledgements
This work has been supported by the grant VEGA No. 2/0070/2009 and bythe Slovak CERN Commission. The author also thanks Pierre Grang´e for fruitfull discussions.
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