Gupta, Sen N D

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ON A REPRESENTATION OF THE DIRAC MATRICESN. D. SEN GUPTATata I^tstitute op pondamental E bseaboh,OoLABA, B ombay-5.(Jieoeivcd October 3, 1966)ABSTRACT. Tlio object of the j)ttpor is t > constract a representation of the Dirac matrices adaptable to tlie free particle spinor.I N T R t ) D t r C T I O NIn order to obtain explicitly the solutions of the Dirac equation and to study their properties, it is the usual practice to start with suitable representation of the Dirac matrices, satisfying the anti-commutation relations. Conversely, one can ask the question whether one can build up a representation of the Dirac matrices adapted to the solution of the Dirac equation for a particular problem. The ob­ject of this short paper is to construct a representation of the matrices, in the spinor space of free particle. The construction of the matrices, in this way, is very instructive as it gives bettor insight to their characteristic properties related to the splitting up and mixing of the positive and negative energy states, as well as those of the spin states. The process is also interesting as it shows clearly where is the liberty of the choice and how the elements are determined. Finally, it also exhibits the natural role of the Pauli matrices in the representation and their invariance properties, which have been recently studied by the author (1965).In the next section, the representation of the matrices in the space of free particle spinor has been developed. In the last section some of their properties, e.g. transposition operation, are discussed.We will need frequently the Pauli matrices which wo take as47cr, = 0. 1 0, —i 1, 0, CTi = > =1, 0 i, 0 0, - 1The very nature of the problem proposed will lead us to hermitian matrices. However, we do not loose any generality, as it is well-known tha t any non- hermitian representation may bo made equivalent to a hermitian one by suitable similarity transformation (Pauli (1936)).—> ^  ^  fFor any space vector q we will use the notation to denote (a. q )l\q \\  so tha t =  1. over a matrix denotes its hermitian conjugate.278R E P R E S E N T A T I O N  OP T H E  M A T R I C E SLet the free particle spinors be U^{p) and ?7_(p) corresponding to the positive and negative energies respectively. HenceOn a Bepresentation of the Dirac Matrices 379{(a. p)+^mc}U+(p) =  -PoU+{p) {(«• P)-\-firnc}U_(p) =  PoU4p)* } ... (1)where p^ =  +{(<*• Let us introduoe the matrices a+ and a_ definedby1a . Po {(a. p)+^mc}andsuch that anda _ = -  {mctp—ftlp ] )  Poa+® — 1, a_® =  1a+«_+a_a+ =  0.... (2)... (3)... (4) ... (5)— f  — rLet j  and k be two mutually orthogonal unit vectors in the plane perpendicular —►to p, i.e.. ^ ^ ^ ^ y{p.j) =  0, { p .k ) ~ 0  and (j. ifc) =  0. ... (6)— y  — ►I t  is clear that j, k arc arbitrary up to a rotation in their plane.Since is an involution and traceless one, it has two distinct pairs of eigen­vectors with eigen-values t-1, —1. Lot r/^.^and be tw'o mutually orthogonal eigen-vectors of a.^ . with eigen-values + 1  and similarly U J  and be those with eigen-values —1. These four vectors are mutually orthogonal and form the basis of our representation. Clearly in this basis (X^  is a diagonal matrix with elements (4“1> +1> —L —1).The matrices which commute with leave the subspaccs of positive and negative energy states invariant. We can take any of the matrices which anti­commutes with a„,. to connect the positive energy states with those of negative energy and vice versa. In order that the representation is symmetric, with res- pect to positive and negative energy states, we take as the matrix which just connects positive and negative energy states (without changing spin states) and conversely, as it is an involution. So that,and (5 =  1. 2) (7)Thus a+ and a_ are given bye, 0 0, eand a_ =0, —c e, 0where c is the 2 x 2  unit matrix.Next let us consider the operator which interchanges only the spin states with same energy. The matrix S, for such an operator, is380 N. D. Sen Ghipta«4 (8)00, 0-1... (9)where cTi is the Pauli matrix given above. I t  clearly commutes with a+ and a_. The most general matrix which commutes with both and a_ is a linear combina­tion ofajCtk, cc+<x,jxie, cx.+iX’-.ocj.Again for symmetry, let us take S to be proportional to ajt so tha tS =  iocfCk.In  order to find ocp a* separately we note that a + U jU — —(XjU+• and Thus the matrix Xj is of the form0, crO', 0(10)X )  = (11)where v  is any 2 x 2  matrix. I t  is easy to see tha t a  should satisfy the following conditionsanda  O’ =  c, c + tr  =  0 1or a-i+Xi (T — 0, Jsuch that xj^ — 1, and «/ anti-commutes with a_ and x+. Hence<r =  i(cos 6 (Tj4-Bin & o^ a).where $ is any real parameter. Hence from Eq. (10), it follows that0, </Xk =10withso thatr', 0o' =  i(sin 9 (Tj —cos 6 <r^  <r<r'4-<r'o’ =  0. }(12)(13)(14)(15)As mentioned above j  and k are indeterminate but for a rotation in the ( j  — ib) plane, thus we can choose 0, in Eqs. (13) and (15), to bo zero so thatOn a Representation of the Dirac Matrices 3810, cTa 0, —0-8; a* == f—<ra 0 (T3, 0Finally the fifth mutually anti-commuting matrix0, —0*1cc^ OLjXjOLjc =  — ia+a_S =  i0(16)(17)which is a matrix with only non-vanishing anti-diagonal elements.D I S C U S S I O l f  SThe construction of the matrices shows hoW the Pauli matrices enter in the expression of the Dirac matrices in an obvious manner. The typical block character of all the four a_, ay, ocic, (x.j^ (x>^(XjCx,ic is evident from their anti-commuting properties with oc^ , which is diagonal. All these four matrices connect positive energy states to the negative energy and vice versa. I t  is interesting to note thattUjOCkand0*1, 0 0, 0-1—ia+a_ajfc = 0'2> 00, 0-3(18)They are isomorphic to the Pauli matrices. I t  may be pointed out that these are also the only three linearly independent matrices which commute with and a_. Thus our choice of S ~  iocjajcy i^ Eq. (10), is nothing but a choice of a particular representation of Pauli matrices. Any other choice would amount to a 3-dimen- sional real orthogonal rotation among these three matrices.The representation constructed here is slightly different from those usually referred (Dirac (1928), Kronig (1938)). In this representation, a^, J3 (obtained from a+ and a_) and ay are real; ajc and oL^ C^LjOLk are purely imaginary. Thus oLk and are anti-symmetric and aj,, /?, and ay are symmetric. The matrixB  (Pauli (1936), which induces the transposition, is This representationis one of the second category of representation as noted by the author (1966).A C K N O W L E D G M E N TThe author acknowledges the support of the Atomic Energy Commission, Government of India.R E F E R E N C E SDirac, P. A. M., 1928, Proo. Roy Soo. 117, 610. Krctxners, H. A,, 1937, Proo, Aoad> AmsL 40, 814. Pauli, W., 1936, Ann. de Vlnet. H, Poinoare 6, 109. Sen Gupta, N. D., 1965, II. Nuo. 04m. 86, 1181.

On a Representation of the Dirac Matrices

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On a Representation of the Dirac Matrices

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