Coherent states, vacuum structure and infinite component relativistic wave equations
aa r X i v : . [ qu a n t - ph ] O c t Coherent states, vacuum structure and infinite componentrelativistic wave equations
Diego Julio Cirilo-Lombardo
National Institute of Plasma Physics (INFIP),Consejo Nacional de Investigaciones Cientificas y Tecnicas (CONICET),Facultad de Ciencias Exactas y Naturales, Universidad de Buenos Aires,Ciudad Universitaria, Buenos Aires 1428, Argentina andBogoliubov Laboratory of Theoretical Physics,141980, Joint Institute for Nuclear Research,Dubna (Moscow Region), Russian Federation (Date textdate; Received textdate; Revised textdate; Accepted textdate; Published textdate)
Abstract
It is commonly claimed in the recent literature that certain solutions to wave equations of positiveenergy of Dirac-type with internal variables are characterized by a non-thermal spectrum. As partof that statement, it was said that the transformations and symmetries involved in equationsof such type correspond to a particular representation of the Lorentz group. In this paper wegive the general solution to this problem emphasizing the interplay between the group structure,the corresponding algebra and the physical spectrum. This analysis is completed with a strongdiscussion and proving that: i) the physical states are represented by coherent states; ii) thesolutions in previous references [1] are not general, ii) the symmetries of the considered physicalsystem in [1] (equations and geometry) do not correspond to the Lorentz group but to the fourthcovering: the Metaplectic group
M p ( n ). ontents I. Introduction and results
II. Relation with the squeezed vacuum
III. The solution
IV. Bargmann representation: analytical vs. geometrical viewpoint
M p (2) generalized coherent states in the Bargmann representation 108
V. The Limit ǫ → m VI. Complete equivalence between Sannikov’s representation andMetaplectic one
VII. Concluding remarks
VIII. Acknowledgements
IX. References
I. INTRODUCTION AND RESULTS
For the last 50 years, there has been an increase in the interest with respect to twofundamental points of theoretical physics: the new representations of algebras with variablesof the harmonic oscillator and the study of relativistic wave equations. These two pointswere developed placing great attention on the condition of positive energy and the role ofthe spin in these representations. The main motives were the theoretical problems of optics,the positive energy spectrum of physical states and the close relation between the spin andthe generalized statistics. Despite the recent interest and the continuous efforts into thestudy on compact groups and their relationship to physics, there was no major progress onthe issue.For example, in a recent reference it was claimed that certain solutions to wave equationsof positive energy of the Dirac type with internal variables have as the main characteristic a102on-thermal spectrum. As part of that statement, it was said that the transformations andsymmetries involved in a such an equation with correspond to a particular representation ofthe Lorentz group.In this work we will demonstrate that both claims and the same state solutions in [1] areunfortunately not fully correct. Calculating the solutions of the physical system of [1] weshow that:i) the solutions are coherent states, as described before (e.g.[2,5,6,7,8,9]);ii) we show that the transformations and symmetries involved into the relativistic waveequation of [1] do not belong to the group of Lorentz but to the double cover of thegroups Sp (2) and SU (1 , M p (2) [2,6,7];iii) and that these solutions have, in general, a thermal spectrum going under certainconditions, to the non-classical behaviour (squeezed)[1,5].Regarding the theoretical basis of the problem we start as follows:Let such a spinor such that can be described schematically by the chain: A α : ∈ M p (2) ⊃ Sp (2 R ) ∼ SU (1 , ⊃ SO (1 , ≈ L (3) (1)(take not of the above structure that will be important into the analysis that follows) thatis defined as: A α = aa + β ⇒ [ A α , A β ] = ǫ αβ (2)where a and a + are standard annihilation and creation operators, respectively. As we willsee soon, there exists a close relation with the squeezed vacuum structure. The equationto solve has the typical structure of the positive energy equation with internal variables, asproposed by Majorana[4] and Dirac[5], and is explicitly written as (cid:0) σ i ∂ i − m (cid:1) βα A β | ψ i = 0 (3)In reference [1], similarly to the case of the Dirac positive energy equation, a wave solutionwas proposed as: | ψ i = e ip · x | u i (4)The first wrong fact in ref.[1] is to assume a priori that the momentum p and x in theexponent of the proposed wave equation (4) commute with the annihilation and creation103perators a and a + . Consequently, in our analysis we will to consider the phase spacecoordinates p and x in the exponent of the proposed wave equations as constants or as ifthe annihilation and creation operators a and a + act in an internal or auxiliary space.Only under these conditions we can insert (4) in (3) obtaining:( ip i σ i − m ) βα aa + β | u i = 0 (5) ip − m ip − p ip + p − ip − m aa + β | u i = 0 (6)At this point the second wrong fact in [1] is evident: the author remains with only onecomponent of the spinor solution. In fact if we impose the same conditions as in [1] namely p i = (0 , p, iε ) we have ε + m p − p m − ε aa + β | u i = 0 (7) ( ε + m ) a + pa + − pa + ( m − ε ) a + β | u i = 0 (8)Notice that there are two different and simultaneous conditions that | u i must satisfy. If weput now p = 0 as in the ref.[1] then ( ε + m ) a ( m − ε ) a + β | u i = 0 (9)Here we clearly see that | u i cannot be the Fock vacuum | i as stated ref [1] (it can onlybe if m = ε ) . Through the next sections we will find the true vacuum, the spectrum and thesolution of the problem.
II. RELATION WITH THE SQUEEZED VACUUM
Looking at expressions (7,8)it is not difficult to see that these can be obtained fromsimilar form as the squeezed vacuum. The squeezed vacuum is generated by the
M p (2)transformation U = S ( ξ ) A α → S ( ξ ) aa + α S † ( ξ ) = λa + µa + λ ∗ a + + µ ∗ a α (10)104here λ ( ξ ) and µ ( ξ ) satisfy | λ | + | µ | = 1 , e.g. SU(1,1) elements.We must note that the RHS of eq. (10) is governed by the operators S ( ξ ) ∈ M p (2) beingthe right side affected by a matrix representation of SU (1 ,
1) as follows S ( ξ ) aa + α S † ( ξ ) = λ µµ ∗ λ ∗ aa + α (11)Clearly the above equivalence is only local (infinitesimal) since at the level of the groupstructure (see the chain (1)) there is an homomorphism relationship. The homomorphismsbetween M p (2) and SU (1 ,
1) (or Sp (2 R )) , which are two to one and four to one in the caseof SO (1 , α (polar)-parameterization [6,7] in the usual way: S ( α ⊥ , α ) ∈ M p (2) → s ( α ⊥ , [ α ] π ) ∈ Sp (2 R ) (12) → s ( α ⊥ , [ α ] π ) ∈ SU (1 ,
1) (13)[ α ] π ∈ ( − π, π ] → [ α ] π mod 4 π, α ⊥ ∈ R , α ∈ ( − π, π ] for the SU (1 ,
1) (or Sp (2 R )) case(14) → [ α ] π mod 8 π, α ⊥ ∈ R , α ∈ ( − π, π ] (15)for the SO (1 ,
2) caseConsequently, it is clear that the ”two to one” and the ”two to four” nature are involvedin the reduction of the range of the parameter α . This is the main reason why, for thephysical scenarios of current interest, the above parameterization is better than the Iwasawa(KAN) one.The most general expression for an element of the Metaplectic group can be computed,with the following result: e A ( aa + + a + a ) + Ba +2 + Ca = e − A/ exp (cid:18) Ba +2 ∆ coth ∆ − A (cid:19) exp (cid:20) H ln (cid:18) ∆ sec h ∆∆ − A tanh ∆ (cid:19)(cid:21) exp (cid:18) Ca ∆ coth ∆ − A (cid:19) , ∆ ≡ √ A − BC, H ≡ (cid:18) aa + + a + a (cid:19) , b N ≡ a + a (16)where the Baker-Haussdorf-Campbell formula was used. A, B, C are arbitrary in principleonly linked by expression (16) (all group theoretical propierties of the noncompact groups105nvolved, were assumed there). Therefore, with the parameters as given by expressions (7)to (9) S ( ξ ) takes a concrete form as follows S ( ξ ) = exp (cid:18) pm + ǫ a +2 (cid:19) (cid:18) √ m − ǫ (cid:19) / ( ∞ X n =0 n ! (cid:20) ln (cid:18) √ m − ǫ (cid:19) b N (cid:21) n ) exp (cid:18) − pm − ǫ a (cid:19) , thus, the unitary (squeezed) operator acting on the true vacuum (fiducial vector) definesthe following general state | ξ i ≡ S ( ξ ) | z i (17) III. THE SOLUTION
We arrive to the construction of coherent states on a general vacuum: A | i + B | i with A and B depending on initial and boundary conditions. If | z i ≡ A | i + B | i then | ξ i ≡ S ( ξ ) | z i = exp ( αa +2 )( m − ǫ ) / (cid:20) A | i + B ( m − ǫ ) / | i (cid:21) (18)= ( m − ǫ ) − / ∞ X n =0 α n n ! (cid:0) a +2 (cid:1) n (cid:20) A + B ( m − ǫ ) / a (cid:21) | i , (19) with α ≡ p/ m + ǫ (20)Notice that a annihilates | z i but a does not. After standard normalization, the constantsin the ”thermal” (photon) case reach the critical point that is when the quantum statesolution is simultaneously eigenstate of a and a , they take the particular fashion A = (cid:0)(cid:12)(cid:12) m − ǫ (cid:12)(cid:12) + p sign (cid:0) ǫ − m (cid:1)(cid:1) / , (21) B = (cid:0)(cid:12)(cid:12) m − ǫ (cid:12)(cid:12) + p sign (cid:0) ǫ − m (cid:1)(cid:1) / = A (22)We have a standard coherent state (eigenstate of the operator a ) as a linear combinationof two states belonging to H / and H / respectively (that are independent coherent statesas eigenvalues of a ). In this particular case we have | z i th = A (1 + A a + ) | i (23)notice that this vacuum is not singular at m → ǫ but is analytically continued into thecomplex plane where it is defined: | ξ i th ≡ S ( ξ ) | z i th = (cid:18) p sign ( ǫ − m ) | m − ǫ | (cid:19) / e p/ m + ǫ a +2 " (cid:18) p sign ( ǫ − m ) | m − ǫ | (cid:19) / a + | i (24)106 V. BARGMANN REPRESENTATION: ANALYTICAL VS. GEOMETRICALVIEWPOINTA. The Bargmann representation
We have so far worked mainly with the photon-number description of the Hilbert space H and the operators a, a + . In this section we analyze the misunderstanding pointed outpreviously introducing the Bargmann representation.The Bargmann representation of H associates an entire analytic function f ( z ) of a com-plex variable z , with each vector | ϕ i ∈ H in the following manner: | ϕ i ∈ H → f ( z ) = ∞ X n =0 h n | ϕ i z n √ n ! , (25) h ϕ | ϕ i ≡ || ϕ || = ∞ X n =0 |h n | ϕ i| (26)= Z d zπ e −| z | | f ( z ) | , (27)where the integration is over the entire complex plane. The above association can be com-pactly written in terms of the normalized coherent states of the Barut-Girardello type,namely, (right) eigenstates of the annihilation operator a : a | z i = z | z i , (28) | z i = e −| z | / ∞ X n =0 z n √ n ! | n i , (29) h z ′ | z i = e ( −| z | / −| z ′ | / z ′∗ z ) (30)then, we have f ( z ) = e −| z | / h z ∗ | ϕ i (31)However, f ( z ) must be as | z | → ∞ so that || ϕ || is finite. In this particular representation,the actions of a and a + , and the functions representing | n i are as follows: (cid:0) a + f (cid:1) ( z ) = zf ( z ) , (32)( af ) ( z ) = df ( z ) dz , (33) | n i → z n √ n ! (34)107 . M p (2) generalized coherent states in the Bargmann representation
Having introduced the necessary ingredients, we can now describe the physical states ofthe system under consideration.i) The H / states occupy the sector even of the full Hilbert space H and we may describethem as follows f (+) ( z, ω ) = (cid:0) − | ω | (cid:1) / e ωz / (35)= (cid:0) − | ω | (cid:1) / X m =0 , , ,.. ( ω/ m m ! z m (36)then, in the vector representation we have: (cid:12)(cid:12) Ψ (+) ( ω ) (cid:11) = (cid:0) − | ω | (cid:1) / X m =0 , , ,.. ( ω/ m m ! √ m ! | m i (37)consequently, the number representation is obtained as: h m (cid:12)(cid:12) Ψ (+) ( ω ) (cid:11) = (cid:0) − | ω | (cid:1) / ( ω/ m m ! √ m ! (38) h m + 1 (cid:12)(cid:12) Ψ (+) ( ω ) (cid:11) ≡ H / states occupy the odd sector of the full Hilbert space H and we may describethem as before: f ( − ) ( z, ω ) = (cid:0) − | ω | (cid:1) / ze ωz / (39)= (cid:0) − | ω | (cid:1) / X m =0 , , ,.. ( ω/ m m ! z m +1 (40)then, in vector representation we have: (cid:12)(cid:12) Ψ ( − ) ( ω ) (cid:11) = (cid:0) − | ω | (cid:1) / X m =0 , , ,.. ( ω/ m m ! p (2 m + 1)! | m + 1 i (41)The number representation is consequently: h m + 1 (cid:12)(cid:12) Ψ ( − ) ( ω ) (cid:11) = (cid:0) − | ω | (cid:1) / ( ω/ m m ! p (2 m + 1)! (42) h m (cid:12)(cid:12) Ψ ( − ) ( ω ) (cid:11) ≡ H = H / ⊕H / , is trivially describedas follows f ( z, ω ) = f (+) ( z, ω ) + f ( − ) ( z, ω ) (44)= (cid:0) − | ω | (cid:1) / X m =0 , , ,.. ( ω/ m m ! z m h (cid:0) − | ω | (cid:1) / z i (45)Then, in complete analogy as their even and odd subspaces, the corresponding states aredescribed by:Ψ ( ω ) = Ψ (+) ( ω ) + Ψ ( − ) ( ω ) (46)= (cid:0) − | ω | (cid:1) / X m =0 , , ,.. ( ω/ m m ! √ m ! h (cid:0) − | ω | (cid:1) / a + i | m i (47) h m | Ψ ( ω ) i (cid:0) − | ω | (cid:1) / ω/ m m ! √ m ! , ( m even) (cid:0) − | ω | (cid:1) / ω/ m m ! p (2 m + 1)! , ( m odd) (48)where the link between the physical observables and the group parameters is given by thefollowing expression (measure): (cid:18) p sign ( ǫ − m ) | m − ǫ | (cid:19) / → (cid:0) − | ω | (cid:1) / (49) V. THE LIMIT ǫ → m This is precisely the limit | ω | → H ). What happens is that in thelimit ǫ → m the density of states corresponding to H / is greater than that of the oddstates belonging to H / . It is for this reason that states belonging to H / , will survive inthis limit. As we will see in a separate publication, that there is a particular case of thetwo-dimensional electron transport with a magnetic field in the plane whose states belongto metaplectic group. 109 I. COMPLETE EQUIVALENCE BETWEEN SANNIKOV’S REPRESENTA-TION AND METAPLECTIC ONE
The main characteristics of the particular representation introduced in [2] is the followingcommutation relation that defines the generators L i :[ L i , a α ] = 12 a β ( σ i ) αβ (50)The above representation which corresponds to a non-compact Lie algebra with the followingmatrix form [5,7]is: σ i = i , σ j = − , σ k = − , (51,52,53)that fulfils evidently: σ i ∧ σ j = − iσ k (54) σ k ∧ σ i = iσ j (55) σ j ∧ σ k = iσ i (56)The equivalence that we want to remark is manifested by the following: Proposition 1 the generators in the representation of [2] fulfil: L i = 12 a β ( σ i ) αβ a α = T i (57) where T i are the Metaplectic generators namely[6,7]: T = i (cid:0) a +2 − a (cid:1) (58) T = − (cid:0) a +2 + a (cid:1) (59) T = − (cid:0) aa + + a + a (cid:1) (60) Proof.
Explicitly in matrix form we can write the generators proposed in the paper [2] (andfor instance in[1]) as L i = u M i v (61)110 ≡ (cid:16) a + a (cid:17) (62) v ≡ aa + (63)In the representation (50), that is faithful, and taking into account that σ k enter as ”met-ric”in the sense given by Sannikov[2] we have M = i − = 14 σ k σ i (64) M = − = − σ k σ j (65) M = − = − σ k (66)consequently and by inspection (50) coincides with (61): thus, the equivalence (57) is proved. VII. CONCLUDING REMARKS
In this paper, we have studied from the physical and group-theoretical point of view,the close relation between the Metaplectic group, the Lorentz group and its covering the SL (2 , C ) ones. The main emphasis was put to clarify the existent confusion between therepresentations of the considered non-compact groups. To this end, using a typical exam-ple, a recently posed problem in [1], we solved exactly the corresponding equations to thephysical scenario given in [1] ,highlighting consequently the common errors and misunder-standings that appear to confuse representations: namely, the Metaplectic one with theother non-compact (Lorentz and Special Linear). The analysis was made easier using thegroup generators written with the Harmonic oscillator variables, arriving at the followingconclusions and results:i) the solutions are coherent states, coinciding with previous theoretical descriptions(e.g.[7,8,9]);ii) the transformations and symmetries involved in the equation of [1] do not belong tothe group of Lorentz but to the double cover of Sp (2) and SU (1 , VIII. ACKNOWLEDGEMENTS
I am very grateful to the CONICET-Argentina and also to the BLTP-JINR Directoratefor their hospitality and finnantial support for part of this work.