arXiv:hep-th/0104182v1 23 Apr 2001 The one example of Lorentz group Leonid D. Lantsman, Wissenschaftliche Gesellschaft ZWST, Judische Gemeinde zu Rostock, Wilhelm-K¨ ulz Platz,6. 18055 Rostock. e-mail:

[email protected] Abstract The aim of this work is to show, on the example of the behaviour of the spinless charged particle in the homogeneous electric fi eld ,that one can quantized the velocity of particle by the special gauge fi xation. The work gives also the some information about the theory of second quantisation in the space of Hilbert-Fock and the theory of projectors in the Hilbert space. One consider in Appendix the theory of the spinless charged particle in the homogeneous addiabatical changed electrical fi eld. 1.Introduction The history of physics in xx century is determined with the two great discoveries. This is the A. Einstein’s relativity theory (the special and the general relativity) and the quantum mechan- ics.The hypothesis about the discrete nature of light,which was suggested by M.K.E.L.Planck in 1900y.(as a ’saving’ from the ultra-violet catastrophe:the endless character of the energy density of the eradiation spectrum because of the contribution of the ultra-violet part of the spectrum ) led M.Planck ( in fact against his ’classical education’ as a physicist of xix century with its Maxwell electrodynamics and Newton-Lagrange mechanics) to the conception of the quantum of minimal action ,i.e. to the Planck constant ˉ h .This was a greatest drama of doctor’s Planck life: he could not accept up to his death in 1947 y. the ’quantum revolution’. What is this- the quantum of minimal action? Our reader ,which acquainted with classical La- grange mechanics, is knows that [1] one can represent the phase space Γ as a Γ = Z dq1.dqsdp1.dps(1) where qi,piare the canonical co-ordinates and the canonical momentum correspondingly. The M.Planck’s hypothesis, in the terms of formula (1),signifi es that the cell of the phase space with the size Γ ≤ (2πˉ h)3is not exists ! This conclusion(at least for the quasiclassical approach in quantum mechanics) signifi es non other then Heisenberg inequality-the fundamental low of quantum me- chanics!So,on author’s of this article opinion,an opportunity for the quantum mechanics appear- ance was as long ago in 1900 y.(with account of the accumulated knowledge in geometrical optics :the Fermat principle,-the low of minimal action in geometrical optics, and the optical-mechanical analogy[2].In fact,one can interpret geometrical optic as a quasiclassic theory of photon !) Albert Einstein as it well known also could not accept quantum mechanics (by the of many years friendship with N.Bohr. Their discussions, the examples and the counterexamples ’for’ and ’against’ of the quantum theory,-all this had often the highly stormy nature!) Indeed,the ’precipice’ between these two great theories is not large.There are (quantum mechan- ics and the relativity theory)- the two corner-stones,on whichis based the majestic building of modern theoretical physics. One of examples -the quantum -fi eld theories:quantum electrody- namics,QCD, the electroweak theory.They all are the relativistic quantum theories. The basic postulates of the quantum -fi eld theory,- Wigthman axioms [3] , are based on Lorentz- Poincare group in Minkowski space .The demand of Poincare invariantness of Wigthman func- tions(the vacuum averages of the operator product of some quantum-fi elds), i.e.the demand of relativistic invariantness of S- matrix and the eff ective section; the microcause principle,i.e. zero value of the fi elds (anti)commutator at the space-like interval between them ,- there are the basic displays of special relativity in the quantum -fi eld theory. The cause of such large diff erence between quantum mechanics and ’old’ Lagrange-Newton (clas- sical) mechanics is the Planck constant ˉ h. The introduction of the quantum ˉ h in theoretical physics lead to the replacement of Poisson bracketsonto the commutation relation between the canonical co-ordinates and the canonical momentum. These canonical comutation relations turn 1 into Poisson brackets of Hamilton mechanics in the limit of ˉ h=0 [4]. The construction of the eigenfunctions of the quantum operators (i.e. of the vectors of Hilbert space):in par- ticular for the quantum-fi eld operators, which we consider as a canonical co-ordinates or a canonical momentum; the solution of the movement equations (Schrodinger equation in non-relativistic quantum mechanics or the quantum-fi eld equations (Dirac equation,Klein-Gordan-Fock equation, Rarita-Schwinger equation,- are the basic examples of the relativistic equations),- are the basic problems of the quantum theory). This is very important that the quantum- fi eld operators have the dimension of the canonical co-ordinates or the canonical momentum. One writes down the movement equations, issuing from the classical Lagrangian of correspond- ing particle and the Lagrange equations as a conditions of the minimal action (for example- the case of the scalar particle).Then we replace all classical momenta on the momentum operators ?iˉ h?/?xi . One calls all this process as the fi rst quantization (it is appropriate to mention here that the conclusion of Dirac equation is based on Schrodinger equation and on the superposition principle ( the basic principle of quantum mechanics) which dictates the form of energy operator (the quantum-mechanical generalization of Hamiltonian):this operator is proportional to Dirac matrixes[5].Thus we obtain this equation from the quantum-mechanical principles. But we can conclude it as Lagrange equation [6] ). The second quantization this is ([7],[3]) the point of view onto the quantum fi eld as a multi-particle fi eld .The every particle we then interpret as a one degree of freedom . Then we construct Hilbert- Fock space of second quantization. The form of the eigenvectors of Hilbert-Fock space is given, for example, in monograph [3] (look formula (7.99) ).This form depends on helisity s (explicity and through the spinors ωαi. On the language of Penrose fl ag structure of the space-time, the spinors ωαiare [8] the main spinors ,by which we decompose arbitrary spin-tensor ψ(ω,?) = P α1.α2j;β1′.β2k′ωα1.ωα2j?β1 ′.?β2k′ (2) Thus formula (7.99) in[3] is in fact the usual decomposition of the wave function on the space and the spinor parts ( in the momentum representation) The structure of Hilbert space is determined with[3] Gelfand-Naimark-Segal (GNS)construction. Let some algebra with involution U is given. Let us denote this algebra as C*.Then exists the isomorphism π of algebra U into algebra B (H) of all linear limited operators in the Hilbert space.The representation π is called unreducible if every closed subspace in H, which is invariant relatively to all operators π(A);A ∈ U is ? or all H.The vector Φ ∈ H is called the cyclical vector for representation π if all vectors of the form π(A)Φ where A ∈ U form the total set in H( correspondingly , the such representation with the cyclical vector is called cyclical). If Φ is the vector in H then it generates the positive functional FΦ= (3) on U (in the terms of the probability theory it is the mathematical expectation of the value π(A) in state Φ ).This functional is called the vector functional , associated with representation π and vector Φ. 2 In these terms GNS construction consists in following: One can determine some (cyclical) representation πFof algebra U in the Hilbert space with cycli- cal vector ΦFfor given positive functional F such that F(A) = (4) The representation πFis determined with these conditions unique with precision of the unitary equivalence. The one of the most important for the quantum-fi eld theory are birth and annihilation operators. As it is well known, the spin of the particle determines the two kinds of statistics : the Bose- Einstein statistics for the integer spin and the Fermi statistics for the semi-integer spin (the Maxwell statistics is the limit of these statistics by ˉ h → 0 ). This two kinds of statistics de- termine the two kinds of the commutation relations between the birth-annihilation operators ( we must consider the commutation relations for the Bose statistics and the anticommutation relations for the Fermi statistics). The every physicist-theorist ought to known these relations ,therefore one can omit they here. But following is most important: the action of the birth and the annihilation operators onto the cyclical vector ( in theoretical physics the cyclical vector is called the vacuum vector) .As it is usual in physics, let us denote the vacuum vector as |0 (the ’bra’ vector). Then the action of the annihilation and the birth operators on |0 is expressed as a(Φ)|0 = 0a ? (Φ)|0 = Φ(0.1) (5) for the birth operators a?(Φ) and the annihilation operators a(Φ). The relation (2) and formula (7.99) in [3] show us that one can interpret the vectors of the Hilbert- Fock space as a tensor products: there are the tensor functions of such number of momentum ,how much the particles we consider in our model.But one can decompose the every tensor onto the symmetrical and the antisymetrical parts. And what is more:as we know, the statistics determine the form of this tensor [4]: some symmetrical wave function for Bose statistics and some antisymmetrical wave function for Fermi statistics (it is in fact [8] the consequence of fl ag structure and spinor algebra,based on this structure). If we denote the one-particle Hilbert space as a Λ1, then one can determine by the natural way the multiparticle space as Λ?n 1 ,- the nth(anti)symmetrical tensor power of the Hilbert space Λ1,which we will denote as Λ∨n 1 for Bose statistics and Λ∧n 1 for Fermi statistics. As a particular case, by n=0 we have the scalar fi eld with the usual scalar product Φ0Ψ0 The vectors of the states with diff erent numbers of particles form the mutually-orthogonal sub- spaces in the complete Hilbert space. Hence, it should to introduce the direct sum of the n-particle subspaces.The direct sum L∞ n=0Λ ?n 1 (6) is called the tensor algebra over the Hilbert space Λ1. The vectors of this space are the arbitrary sequences {Φn}∞ n=0 such that Φn∈ Λ?n 1 and ||Φ||2= P∞ n=0 (8) It is obvious that one can identify Λ?n 1 with the n-particle space in above direct sum. If k 6= n then Φk= 0. In particular,the one-dimensional 0-particle space Λ?0 1 is called the vacuum space. It is ’pulled’ onto the vector Φ0≡ |0 with its components (Φ0)0= 1;(Φ0)n= 1 by n6= 0 (9) which is called the vacuum vector. We can consider Φnas a projection of the sequence Φ on Λ?n 1 . The vector Φ is called the fi nite vector, if it hasthe fi nite number of its projections Φn diff erent from zero.It is obvious that the fi nite vectors form the linear manifold which is densein the direct sum (6). The direct sum (6) is divided with the natural way on the direct sum it of its Bose and Fermi components, which we will denote as F∨(Λ1) and F∧(Λ1) correspondingly.These spaces are called the Fock spaces of bosons and fermions correspondingly. The fundamental quantum-mechanical characteristics of the particle are its mass and its spin.We shall consider the one-particle space Λ1as a ’supplied’ with these characteristics, and shall denote it as Λ(m,s). This space is transformed by the unitary representation of Poincare group ( latter acts as an automorphism on Λ(m,s)). One can,of course, consider the spaces Λ∨n 1 and Λ∧n 1 as an eigen-subspaces of the (Hermit) oper- ator of the numbers of particlesN, which is determined on the (fi nite) vectors by formula (NΦ)n= nΦn(10) The construction of Fock space F∨(Λ1) or F∧(Λ1) by given space Λ(m,s)is called the second quan- tization The birth and the annihilation operators allow the interpretation in the terms of the number of particles. Let us determine the symmetrical tensor product of vectors Φ,Ψ of space F∨(Λ1) which is associative,distributive and commutative.And analogous we determine the antisymmetrical ten- sor product of vectors Φ,Ψ of space F∧(Λ1) which is distributive and associative only. Φ ∨ Ψ = Sym Φ N Ψ Φ ∧ Ψ = Antisym Φ N Ψ (11) Let us fi x some vector Φ in formula (11) and let us consider then the maps Ψ → √NΦ ∨ Ψ (12a) or Ψ → √NΦ ∧ Ψ (12b) The both maps are some linear operators on the fi nite vectors Φ of Fock (boson,fermion) spaces. The basic feature of these maps is that the n- particle vector turns into the (n+1)-particle vector , therefore it is called the birth operator of the particle with wave function Φ and denoted a?(Φ). So, in the boson case the birth operator is determined with formula a?(Φ) = √NΦ ∨ Ψ (13a) and analogous in the fermion case as a?(Φ) = √NΦ ∧ Ψ (13b) Then we can determine the annihilation operator as a operator conjugated to a?(Φ). This oper- ator turns the n- particle vector in the (n-1)particle vector. 4 Such is the briefl y sketch of the second quantization theory. It is important that GNS construc- tion of every Hilbert space Λ(m,s)generates the general structure of the Fock space. Thus there exist the two interdependent approaches to the quantization problem : the fi rst quan- tization,which is connected with replacement of Poisson brackets between canonical co-ordinates and canonical momentas onto the commutation relations with Planck consant ˉ h. The second ap- proach, the second quantization, is connected, as we just saw, with the structure of Fock- Hilbert space, with the two statistics and GNS construction. The aim of this article is to show a some original variant of quantization: the quantization of the particle velocity. In fact, we will obtain discrete Lorentz groupe. This is one of examples of quantum groups, interest to which is very large now. Mathematician V.G.Drinfel’d from Kharkow,apparently, is a pioneer in this sphere [9]. One can also recommend among the interesting works of this direction the work [10] of G.W.Delius,[11] of P. Schupp, [12] of M. Vybornov and many other on this theme. Author wants to dedicate this work to respectful memory of his fi rst teacher in theoretical physics- doctor V.M. Pyg, whichdeceased sudden in 1998 y. He was scientist ,well known with his works in the sphere of functional integration, conformnal gravitation, fi bre bundles both in Soviet Union and far from its frontiers. The invaluable help to author by the investigations, which preceded this work, rendered the em- ployee of Kharkow low temperature Institute- doctor G.N. Geistrin: the well known Soviet and Ukrainian mathematician. He found the time for me in the diffi cult conditions of the ’postsoviet’ period in the Ukraine; I am very grateful him for this. II.The equation of movement for the scalar particle. Let us [13] consider the charged scalar spinless particle in the electromagnetic fi eld: the one of most simple theories. We set here ˉ h=c=1.Then the movement equation for this particle is Klein- Gordon-Fock equation [( ? ?xm ? ieAm)2? m2]Ψ = 0 (14) Let us choose the 4-potential