aa r X i v : . [ h e p - t h ] A ug Bosonization of Weyl Fermions and Free Electrons
E. C. Marino
Instituto de F´ısica, Universidade Federal do Rio de Janeiro,C.P.68528, Rio de Janeiro RJ, 21941-972, Brazil (Dated: August 18, 2015)The electron, discovered by Thomson by the end of the nineteenth century, was the first exper-imentally observed particle. The Weyl fermion, though theoretically predicted since a long time,was observed in a condensed matter environment in an experiment reported only a few weeks ago.Is there any linking thread connecting the first and the last observed fermion (quasi)particles?The answer is positive. By generalizing the method known as bosonization, the first time in itsfull complete form, for a spacetime with 3+1 dimensions, we are able to show that both electronsand Weyl fermions can be expressed in terms of the same boson field, namely the Kalb-Ramondanti-symmetric tensor gauge field. The bosonized form of the Weyl chiral currents lead to theangle-dependent magneto-conductance behavior observed in these systems.
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I. Introduction
Nature exhibits two completely different classes of par-ticles: fermions and bosons. Apparently all matter is ul-timately composed out of fermion particles, namely, elec-trons, quarks and then, protons, neutrons and so on. Theinteractions among these matter particles, conversely,seem to be mediated only by bosonic particles, suchas photons, gluons and massive vector mesons, whichare the quantum-mechanical manifestation of the gaugefields that intermediate such interactions. Is it possi-ble to bridge the gap separating the fermionic matterconstituents from the bosonic conveyors of their interac-tions? Is it conceivable that a super-unification of matterwith its interactions could be achieved?In this work, we generalize the method of completebosonization (bosonization of the fermion field itself) tofour-dimensional spacetime, thereby demonstrating thatboth Dirac (electrons) and Weyl fermions can be ex-pressed in terms of a 2-tensor, Kalb-Ramond bosonicgauge field. The fermion particles then appear as topo-logical excitations of the gauge field.The result has far reaching consequences. In the con-densed matter framework, our results apply to the Weylfermion quasiparticles, experimentally observed very re-cently by photoemission spectroscopy methods in the
T aAs and
N bAs semimetals [1]. In particle physics, con-versely, it bears on the very nature of elementary pointparticles such as the electron. Take, for instance QED,which results from the interplay of the bosonic gauge vec-tor electromagnetic field with its charge bearing fermionsources such as electrons, for instance. With the full elec-tron bosonization, one would be able to have QED for-mulated as a theory involving just gauge fields describingboth electrons and photons.Bosonization is an extremely powerful method bywhich fermionic fields are mapped into bosonic ones.It was firstly developped in one spatial dimension [2–4], where it found important applications ranging fromcondensed matter to particle physics. The Tomonaga- Luttinger [5] system is an example of the former, whileQED of massless fermions [6] is an example of the lat-ter. The method of bosonization has led, in the firstcase, to the obtainment of the exact energy spectrum,correlations functions, response functions, etc. of suchstrongly interacting nonlinear systems. In the secondcase, conversely, it produced and exact operator solutionof QED , which provided a deep understanding of generalfeatures of QCD , such as color confinement/screening,chiral symmetry breaking, topological vacua, instantonsand the hadronic spectrum within an exactly soluble,completely controlable system.The excellence of the bosonization method has madedesirable its generalization to higher dimensions. Earlyattempts in this direction were made in [7–9]. Later de-velopments can be found in [10–12]. Nevertheless, a com-plete bosonization of fermions, by which we mean the fullexpression, not only of the lagrangean and currents, butof the fermion field itself in terms of a bosonic one, wasachieved for the first time for D >
II. Weyl and Dirac Fermions
A Dirac Fermion decomposes in two-components Weylfermions, such that ψ = (cid:18) ψ L ψ R (cid:19) (1)when we use the Weyl representation of the Dirac ma-trices. The lagrangean of a massless Dirac field, in thiscase, reads L = i ¯ ψγ µ ∂ µ ψ = ψ † L σ µ ∂ µ ψ L + ψ † R ¯ σ µ ∂ µ ψ R (2)where σ µ = ( I , σ i ) and ¯ σ µ = ( I , − σ i ) and I is the rank-2identity matrix.A mass term for the Dirac field is given by L M = M ¯ ψψ = M h ψ † L ψ R + ψ † R ψ L i (3)The current and chiral current, accordingly, will be givenrespectively by¯ ψγ µ ψ = ψ † L σ µ ψ L + ψ † R ¯ σ µ ψ R ¯ ψγ γ µ ψ = ψ † L σ µ ψ L − ψ † R ¯ σ µ ψ R (4) III. Pre-Bosonization
It is convenient to express the energy and momentumin terms of the rapidity variable χ ∈ [0 , ∞ ), such that inthe positive energy, time-like region of Minkowski space,we have (in other regions there will be corresponding ex-pressions) k = k cosh χ ; | k | = k sinh χ (5)where k = p k µ k µ .We then have, σ µ k µ = k [ I cosh χ + ˆ r · σ sinh χ ] (6)where ˆ r is the radial unit vector of the spherical coordi-nate system.Before bosonizing the Weyl fermion fields ψ L and ψ R ,we introduce the new spinor fieldsΨ L = T L ψ L Ψ R = T R ψ R T L,R = h I cosh χ ± ˆ ϕ · σ sinh χ i h I − i ˆ θ · σ i (7)such that ψ † A σ µ k µ ψ A = Ψ † A k I Ψ A ; A = L, R (8) The canonical transformations T L,R , render the mass-less lagrangean diagonal. It is similar to the Foldy-Wouthuysen transformation but is not unitary.From the previous equation, we conclude the Ψ-fieldeuclidean correlation functions are h Ψ L ( x )Ψ † L ( y ) i = h Ψ R ( x )Ψ † R ( y ) i = I π | x − y | h Ψ L ( x )Ψ † R ( y ) i = h Ψ R ( x )Ψ † L ( y ) i = 0 (9)Considering that T † L T R = T † R T L = I , it follows thatthe mass term becomes L M = M ¯ ψψ = M h Ψ † L Ψ R + Ψ † R Ψ L i (10) IV. Lagrangean and Current Bosonization
Consider the generating functional of current correla-tion functions Z [ J ] = Z DψD ¯ ψ exp (cid:26) − Z d z [ i ¯ ψ ∂ψ − J µ j µ ] (cid:27) == Z [ J ] Z N> [ J ] Z [ J ] = exp (cid:26) Z d zJ µ Π µν J ν (cid:27) (11)where Π µν is the one-loop, massless vacuum polarizationtensor and Z N> [ J ] contains only higher-than-quadraticterms. It follows that the fermion current j µ = ¯ ψγ µ ψ two-point correlation function in momentum space is h j µ j ν i ( k ) = Π µν ( k ) = 124 π (cid:2) k δ µν − k µ k ν (cid:3) (12)Then assuming the fermionic current j µ = ¯ ψγ µ ψ is ex-pressed as j µ = 12 K µαβ (cid:2) B Lαβ + B Rαβ (cid:3) in terms of the chiral bosonic 2-tensor fields B L,Rµν , wherethe tensor K µαβ is to be determined, we may write therelevant generating functional as Z [ J ] = Z DB Lµν DB Rµν × exp − Z d z X A = L,R (cid:20) H Aµνα H µναA − J µ K µαβ B Aαβ (cid:21) (13)where H µνα = ∂ µ B να + ∂ ν B αµ + ∂ α B µν . Since it is a freetheory, arbitrary 2 n -point correlation functions of thefields Ψ L,R will be products of (9) and, in order to repro-duce them within the bosonic theory we do not have togo beyond Z [ J ] in order to find the bosonic lagrangeanand current [18].From the expression above, we may infer the followingbosonization formulas for the lagrangean and current i ¯ ψ ∂ψ = X A = L,R H Aµνα H µναA ¯ ψγ µ ψ = 12 r (cid:3) π ǫ µναβ ∂ ν (cid:2) B Lαβ + B Rαβ (cid:3) (14)For the axial current, we have, in the absence of an elec-tromagnetic field¯ ψγ µ γ ψ = 12 r (cid:3) π ǫ µναβ ∂ ν (cid:2) B Lαβ − B Rαβ (cid:3) (15)If there is an applied EM field A µ , however, the axialcurrent will acquire a topological term [19]. For the chi-ral, L, R
Weyl currents, according to (4), we must haveconsequently j µL = ψ † L σ µ ψ L = 12 r (cid:3) π ǫ µναβ ∂ ν B Lαβ + 12 I µ j µR = ψ † R ¯ σ µ ψ R = 12 r (cid:3) π ǫ µναβ ∂ ν B Rαβ − I µ (16)and consequently, for the Dirac chiral current, j µ ¯ ψγ µ γ ψ = 12 r (cid:3) π ǫ µναβ ∂ ν (cid:2) B Lαβ − B Rαβ (cid:3) + I µ (17)In the absence of an external EM field, the I µ topologicalterm just vanishes. When there is an EM backgroundfield A µ , then [19] I µ = 14 π ǫ µναβ A ν ∂ α A β (18)implying that ∂ µ j µ = ∂ µ I µ = − π F µν ˜ F µν (19)where the last term is the Chern-Pontryagin topologi-cal charge density of the EM field, namely, the chiralanomaly [20]. V. Field Bosonization
We now consider the bosonization of the fields Ψ
L,R .For this purpose, the fundamental step is the identifica-tion of the underlying duality structure and the relevantdual operators. The basic building blocks were intro-duced in [23]: µ ( x ) = exp (cid:26) ia Z x −∞ dξ µ (cid:3) − / ǫ µναβ ∂ ν B αβ ( ξ ) (cid:27) . (20)and σ ( S ( C y )) = exp ( − i b πρ Z S ( C y ) d ξ µν B µν ( ξ ) ) . (21) where a and b are dimensionless real parameters and theKalb-Ramond field B µν is either L or R . S ( C y ) is asurface having as boundary a circle of radius ρ , centeredat the point y .The µ operator creates quantum eigenstates of thetopological charge of the Kalb-Ramond field, which ac-cording to (14), is proportional to the fermion charge.The S ( C y ) operator is nonlocal, in the sense it dependson a closed curve C y , very much like the Wilson loop.It creates a quantum state bearing a flux of the (vector)source of the Kalb-Ramond field, Q i = ∂ j H ij , alongthat closed curve.Since we are bosonizing a local field, however, we musttake the local limit, where the curve C y shrinks to apoint. Hence, in order to have a nontrivial result, wemust consider the flux per unit length, by dividing the b coefficient by the string length. Thus the curve C y is as-sumed to be a circle of infinitesimal radius ρ . Notice that a and b are dimensionless parameters to be determined.Let us evaluate now the four-points mixed order-disorder correlation function in the framework of thebosonic Kalb-Ramond theory given by (14). This isquadractic and the order and disorder operators are ex-ponentials of linear forms in the 2-tensor field. Hence, thefunctional integral leading to the correlation function canbe straightforwardly performed, giving the result h σ ( C x ) µ ( x ) µ † ( y ) σ † ( C y ) i =exp (cid:26) − b π [ln µ | x − y | − ln µ | ǫ | ] − a π [ln µ | x − y | − ln µ | ǫ | ] (cid:27) (22)In the limit x → x = x , x → x = x , we obtain h σ ( C x ) µ ( x ) µ † ( y ) σ † ( C y ) i −→ exp (cid:26) − a + b π [ln µ | x − y | − ln µ | ǫ | ] (cid:27) (23)Notice that the infrared regulator µ , completely can-cels in this correlation function. Conversely, for the cor-relator h σµµ † σ i , for instance, the b term would havethe sign of the first logarithm reversed, thus producingan overall ln µ -factor that would force it to vanish. Theinfrared regulator, therefore provides an efficient mech-anism of enforcing the relevant selection rules. The ul-traviolet regulator ǫ , appearing in the unphysical self-interaction terms, conversely, may be removed by renor-malizing, respectively, the operators σ and µ in the cor-relation functions.The natural choice for the bosonization of the Weylfermions, therefore, isΨ L = (cid:18) σµσµ † (cid:19) L Ψ R = (cid:18) σ † µσ † µ † (cid:19) R (24)where the σ, µ operators are expressed, respectively, interms of the chiral tensor fields B L,Rµν . With the choice a + b = 6 π we reproduce the correlation functions(9) completely within the framework of the bosonic fieldtheory.The correlator (23), in particular, would correspond,through bosonization, to h Ψ L ( x )Ψ † L ( y ) i . One can eas-ily verify that the remanining correlation functions in (9)are correctly reproduced by the bosonization formulasabove. Such formulas correspond in D=4 to the Mandel-stam bosonization formula of D=2 [4].Using these bosonization formulas, we can obtain, forinstance, a bosonized version of the Dirac particles massterm (10) that would befit the electron. This would be ageneralization of the sine-Gordon theory.Let us now take the dual operators at a constant time,namely, σ ( S ( C y ) , t ) = exp ( − i b πρ Z S ( C y ) d ξ ij B ij ( ξ, t ) ) .µ ( x , t ) = exp (cid:26) ia Z x −∞ dξ i (cid:3) − / ǫ ijk Π jk ( ξ, t ) (cid:27) . (25)and let us determine their commutation relations. Us-ing canonical equal-time commutation rules for the Kalb-Ramond field and its conjugate momentum Π jk , we ob-tain, for Ψ L, ( x ; C x , t ) = σ ( C x , t ) µ ( x , t ), for instanceΨ L, ( x ; C x , t )Ψ L, ( y ; C y , t ) =exp { iabǫ (2 π − Ω( x ; C y )) } Ψ L, ( y ; C y , t )Ψ L, ( x ; C x , t )(26)where Ω( x ; C y ) is the solid angle comprised by the curve C y with respect to the point x . Since, we want to de-scribe local fields through the bosonization process, asmentioned before, we are going to take the limit wherethe curve shrinks to a point and the solid angle, conse-quently reduces to zero. In this case, the exponential fac-tor above becomes a constant: e iab . The only choice con-sistent with multiple commutations is, then, ab = π .Thisreflects the fact that only fermion or boson local fieldsare allowed in D=4. For the string objects created by σ ( C ), however, the solid angle would not be zero and anarbitrary spin s = ab π would be allowed [23].It is very instructive to investigate how the bosonizedfield behaves under a gauge transformation of the bosonicgauge field, namely B µν → B µν + ∂ µ Λ ν − ∂ ν Λ µ . Weimmediately see that the operator µ is gauge invariant,whereas σ ( S ( C y ) , t ) → σ ( S ( C y ) , t ) exp ( − i b πρ I C y dξ i Λ i ( ξ, t ) ) ≡ σ ( S ( C y ) , t ) e − iϕ ( y ) ϕ ( y ) = b πρ I C y dξ i Λ i ( ξ, t ) (27) We see that a gauge transformation of the bosonic ten-sor field emerges as an U(1) gauge transformation of theWeyl fermions Ψ L and Ψ R . The U(1) gauge transforma-tion of a Dirac field, such as the electron field, for in-stance, according to (24) and (1), would be obtained bysimultaneous gauge transformations of the chiral bosonictensor fields B Lµν and B Lµν with opposite gauge parame-ters Λ Rµ = − Λ Lµ ≡ Λ µ . Then, the U(1) transformation ofa Dirac field is such thatΨ D → e iϕ ( x ) Ψ D ϕ ( x ) = lim ρ → b πρ I C x dξ i Λ i ( ξ, t ) (28) VII. Magneto-Conductance
From (16)-(18) we can infer that, in the presence of anapplied external EM field, the chiral conductivity tensorwill exhibit a term proportional to δ ij (cid:2) | E | | A | − ( E · A ) (cid:3) ∝ δ ij (cid:2) | E | | B | + ( E · B ) − | E | (ˆ r · B ) − | B | (ˆ r · E ) (cid:3) (29)This will account for the magneto-conductance observedin Weyl semimetals [1, 21, 22]. VIII. Concluding Remarks
The bosonization of Dirac and Weyl fields in four-dimensional spacetime, obtained in the present work,opens several new possibilities. Under this new per-spective, one may start to inquire about what ultimatelyare the so-called elementary particles such as electrons,quarks and so on. What are their fundamental attributes,such as charge, spin, color, etc. The bosonization of elec-trons and Weyl fermions, reported here, indicates thatthese elementary fermions are topological excitations inthe framework of a tensor gauge field theory. Electriccharge, for instance becomes the topological charge ofthe bosonic tensor theory.A remarkable connection emerges from our method.As we know, the electromagnetic interaction results fromimposing the invariance under U(1) gauge transforma-tions. As we have seen, however, these became the re-sulting effect of an underlying gauge invariance of thetensor gauge field associated to the particles that carrythe source of the electromagnetic field.Our results seem to indicate that the bosonic ten-sor gauge field is the underlying matrix upon which thefermionic matter fields are created. The properties ofthese fermionic matter particles, such as the way theyacquire mass, for instance, must be deeply influenced bythe subjacent boson field. This may bring some light tothe physics underlying the Higgs particle.This work was partially supported by CNPq andFAPERJ. [1] S. -Y. Xu, et al. , Science 347, 294 (2015); ibid
Science349, 613 (2015); B. Q. Lv et al.et al.