Francesco A. Antonuccio. Department of Theoretical Physics, 1 Keble Road, Oxford OX1 3NP, United Kingdom ABSTRACT

Transcription

1 Towards a Natural Representation of Quantum Theory: I. The Dirac Equation Revisited Francesco A. Antonuccio Department of Theoretical Physics, 1 Keble Road, Oxford OX1 3NP, United Kingdom ABSTRACT An alternative (yet equivalent) formulation of the Dirac equation and corresponding 4-spinor is investigated by considering a representation of the Lorentz Group over a simple non-division algebra. The (commutative) algebra admits a natural operation of `conjugation', and `unitarity' can easily be dened. In contrast to the usual complex representation of the Lorentz group, the one introduced here turns out to be unitary. HEP-TH

2 1 Introduction and Motivation This article is motivated by the observation that the Dirac equation can be concisely formulated without any explicit reference to complex-valued quantities. To see this, consider the following: 1.1 Realising the Dirac Equation If C = is a Dirac 4-spinor satisfying the Dirac equation 1 B@ u 1 +iv 1 u 2 +iv 2 u 3 +iv 3 1C A (1) u 4 +iv 4 m) C =; (2) then the real functions u i ;v i (i=1;:::;4) satisfy the following eight dierential t u 1 x u y v z u 1 + mv 3 t u 3 x u 4 y v 4 z u 3 + mv 1 t v 1 x v 2 y u z v 1 mu 3 t v 3 x v y u 4 z v 3 mu 1 t u 2 x u 1 y v 1 z u 2 + mv 4 t u 4 x u y v z u 4 + mv 2 t v 2 x v y u 1 z v 2 mu 4 t v 4 x v 3 y u z v 4 mu 2 : Curiously, there is a commutative algebra closely related to the real numbers which admits a similarly concise formulation of the equations listed above. Some basic denitions concerning this algebra are presented in the next section. 1.2 The Semi-Complex Number System In this article, we will be considering `numbers' of the form (3) w = t +jx; (4) where t and x are real numbers, and j is a commuting variable satisfying the relation j 2 =1: (5) We call t and x the real and imaginary part of w = t +jx(respectively). Addition, subtraction, and multiplication 2 are dened in the obvious way : (t 1 +jx 1 )(t 2 +jx 2 ) = (t 1 t 2 )+j(x 1 x 2 ); (6) (t 1 +jx 1 )(t 2 +jx 2 ) = (t 1 t 2 +x 1 x 2 )+j(t 1 x 2 +x 1 t 2 ): (7) The zero element is = + j, and two numbers t 1 +jx 1,t 2 +jx 2 are equal if and only if t 1 = t 2 and x 1 = x 2. The set of all such numbers will be denoted by the symbol D, and we shall refer to this set as the semi-complex number system 3 For a more detailed account of this number system, the reader is referred to [1], [2]. 1 The chiral representation for the matrices is assumed here. See [3] for details. 2 One can also easily dene division; see section The more familiar term hyperbolic quasi-real numbers is avoided for the sake of brevity. 2

3 1.3 Dirac's Equation Revisited Dene four (real) 4 4 matrices,,(=;:::;3) by writing = 1 1 ; = ; 2 = 1 1 ; 3 = 3 3 ; (8) 1 1 where 1 =, 1 3 =, and 1 is the 2 2 identity matrix. We choose this 1 notation for later convenience. The anti-commutation relations for the matrices take the form f ; g= 2g ; ; =;:::;3: (9) We now introduce the semi-complex analogue of the Dirac 4-spinor: Let D = B@ a 1 +jb 1 a 2 +jb 2 a 3 +jb 3 1C A ; (1) a 4 +jb 4 where a i ;b i (i=1;:::;4) are real, and suppose D satises the equation m) D =: (11) Separating the real and imaginary parts of this last equation yields a set of eight real partial dierential equations connecting the real functions a i ;b i.interestingly, ifwe make the substitutions a 1! u 2 b 1! v 3 a 2! v 1 b 2! u 4 a 3!u 1 b 3! v 4 a 4!v 2 b 4!u 3 ; (12) then these eight dierential equations become identical to the set of equations listed in (3). In other words, if we make the identication C = B@ u 1 +iv 1 u 2 +iv 2 u 3 +iv 3 1C A $ D = u 4 +iv 4 B@ u 2+jv 3 v 1 ju 4 u 1 jv 4 1C A ; (13) v 2 +ju 3 then equation (2) (Dirac's equation), and equation (11), are entirely equivalent. The suggestion here is that the Dirac equation is not a fundamentally `complex' equation. Fortunately, this observation turns out to be more than just a curious accident, and a better understanding can be obtained by investigating a particular representation of the Lorentz Group. This topic will be taken up next. 3

4 2 The Lorentz Algebra 2.1 A Complex Representation Under Lorentz transformations, the Dirac 4-spinor C transforms as follows: C! e i 2 ( i) e i 2 (+i) C: (14) The six real parameters =( 1 ; 2 ; 3 ) and =( 1 ; 2 ; 3 ) correspond to three generators for spatial rotations, and three for Lorentz boosts respectively. The matrices =( x ; y ; z ) are the well known Pauli matrices. Let us introduce six matrices E i ;F i (i=1;2;3) by writing E 1 = 2 1 x E 2 = i x 2 y E 3 = 1 y 2 z z F 1 = i 2 x x F 2 = 1 2 y y Then transformation (14) may be written as follows: F 3 = i 2 z z (15) C! exp( 1 E 1 2 E E F F F 3 ) C : (16) The algebra of commutation relations for the matrices E i ;F i is given below: (17) [E 1 ;E 2 ]=E 3 [F 1 ;F 2 ]= E 3 [E 1 ;F 2 ]=F 3 [F 1 ;E 2 ]=F 3 [E 2 ;E 3 ]=E 1 [F 2 ;F 3 ]= E 1 [E 2 ;F 3 ]=F 1 [F 2 ;E 3 ]=F 1 [E 3 ;E 1 ]= E 2 [F 3 ;F 1 ]=E 2 [E 3 ;F 1 ]= F 2 [F 3 ;E 1 ]= F 2 All other commutators vanish. Abstractly, these commutation relations dene the Lie algebra of the Lorentz Group O(1; 3), and the complex matrices E i ;F i dened by (15) correspond to a complex representation of this algebra. 2.2 A Semi-Complex Representation It turns out that there exists a semi-complex representation of the Lorentz algebra (17). In order to obtain an explicit presentation, we proceed as follows: Dene three 2 2 matrices =( 1 ; 2 ; 3 )by setting 1 = 1 1 ; 2 = j j ; 3 = 1 1 The reader may like toverify the following commutation relations for the matrices i : : (18) [ 1 ; 2 ]=2j 3 ; [ 2 ; 3 ]=2j 1 ; [ 3 ; 1 ]= 2j 2 : (19) Now redene the matrices E i ;F i (i=1;2;3) by setting E i = 2 j i i ; F i = 1 2 i i 4 ; i =1;2;3: (2)

5 A straightforward calculation shows that these matrices do indeed satisfy the Lorentz algebra dened by the commutation relations (17). Consequently, if D is a semi-complex 4-spinor (as in expression (1)), then under Lorentz transformations, D transforms as follows: D! exp( 1 E 1 2 E E F F F 3 ) D ; (21) where this time, the matrices E i ;F i are semi-complex. We shall discover in the next section that the semi-complex representation has the distinct advantage of being unitary. This means that the exponential in transformation (21) is a unitary matrix. Since the semi-complex algebra admits a very natural operation of `conjugation', the denition of unitarity presented in the next section should look very familiar, and very natural. 3 Unitarity We shall endeavour to present an elementary (i.e. brief) introduction to the semi-complex unitary groups. We begin by dening `conjugation' for semi-complex numbers. 3.1 Conjugation Given any semi-complex number w = t + jx, we dene the conjugate of w, written w, to be w=t jx: Two simple consequences can be immediatetely deduced; for any w 1 ;w 2 2 D,wehave We also have the identity w 1 +w 2 = w 1 +w 2 and (22) w 1 w 2 = w 1 w 2 : (23) w w = t 2 x 2 : (24) Hence w w is real for any semi-complex number w, although unlike the complex case, it may take on negative values. In order to strengthen the analogy between the semi-complex and complex numbers, we often write jwj 2 = w w where jwj 2 is referred to as the `modulus squared' of w. A nice consequence of these denitions can now be stated: For any semi-complex numbers w 1, w 2 2 D, jw 1 w 2 j 2 = jw 1 j 2 jw 2 j 2 : Now observe that if jwj 2 does not vanish, the quantity w 1 = w jwj 2 (25) is a well dened inverse for w. Sowfails to have aninverse if (and only if) jwj 2 = t 2 x 2 =. 5

6 3.2 The Semi-Complex Unitary Groups Hermiticity: Suppose H is a matrix of arbitrary dimensions with semi-complex entries. The adjoint of H, written H y, is obtained by transposing H, and then conjugating each of the entries. We sayhis Hermitian if H y = H, and anti-hermitian if H y = H. For example, the i matrices dened in (18) are Hermitian. Unitarity: The semi-complex unitary group U(n; D) is the set of all n n semi-complex matrices U satisfying the identity U y U =1: (26) The special unitary group SU(n; D) is dened to be the set of all elements U 2U(n; D) with unit determinant: det U =1: (27) Unitarity and Hermiticity: IfHis an n n Hermitian matrix (over D), then e jh is an element of the unitary group U(n; D). Equivalently, e H is unitary if H is anti-hermitian. If the trace of H vanishes, then e jh (or e H in the anti-hermitian case) is contained in the special unitary group SU(n; D). Remark: According to these denitions, the exponential appearing in the Lorentz transformation (21) is an element of the special unitary group SU(4;D), since the traceless generators E i ;F i dened in (2) are anti-hermitian. So the Lorentz group is just a six-dimensional (Lie) subgroup of the fteen dimensional Lie group SU(4;D). 3.3 An Isomorphism: The Spin Group We claim that the semi-complex group SU(2; D) is just the familiar complex group SU(1; 1). In fact, if we restrict our attention to real numbers a 1 ;a 2 ;b 1 ;b 2 satisfying the constraint a 2 1+a 2 2 b 2 1 b 2 2 =1, then the identication a1 +ia 2 b 1 +ib 2 b 1 ib 2 a 1 ia 2 $ a1 +jb 1 a 2 +jb 2 a 2 +jb 2 a 1 jb 1 establishes a (group) isomorphism between SU(1; 1) and SU(2; D), as claimed. The terminology `spin group' for SU(2;D) anticipates a forthcoming article investigating the intimate relationship between this group, spin, and Lorentz invariance in space-time. The identication (28) above suggests that the conformal group SU(2; 2) might be just the group SU(4;D); at any rate, they are both fteen dimensional. Initial investigations suggest that the dierences between these groups might be very subtle. A denitive proof demonstrating the (non)existence of an isomorphism at least between the associated Lie algebras would be an important initial step before investigating the physics of SU(4;D). (28) 6

7 4 The Dirac Equation Revisited II 4.1 Quantisation: A New Perspective The Dirac equation (11) with zero mass (m =)may be written in the form D = H D ; where the operator H turns out to be Hermitian 4 with respect to the semi-complex inner product Z y < D ; D >:= D Dd 3 x: (3) So the scalar quantity j D j 2 := y D D may be viewed as some kind of Lorentz invariant density function 5. So how dowe quantise eld equations in general such as the massless Dirac equation given by expression (29)? Associated with any equation of this form is a propagator K, which, in the path integral formalism, necessitates the evaluation of a path integral K = Z e js D y D : (31) The functional S = S[ y ; ] is an appropriately chosen action, which wemay assume to be realvalued 6. Note that the integrand of this path integral is a semi-complex phase e js, contrasting the usual prescription of integrating over a complex phase e is. Now any semi-complex number may be uniquely decomposed into a real linear combination of the following two projections: p + = 1 2 (1 + j) p = 1 (1 j) (32) 2 (p + and p are projections since p + + p =1,p 2 +=p +,p 2 =p, and p + p = p p + = ). In particular, e js = cosh + j sinh admits the following decomposition: e js = e S p + + e S p : (33) So the path integral (31) takes the form K = K + p + + K p, where the projection coecients K + ;K are given by K + = K = Z Z e S D e S D y D (34) y D (35) If S were positive-denite (e.g. if we replace S with jsj), one would expect (34) to be divergent, although (35) might be well dened. So the original integral (31) may be undened as a whole, but it may nevertheless possess a well dened projection. Mathematically, this is an elegant way of separating unwanted innities. Moreover, the semi-complex formalism appears to have removed the troublesome complex phase in the path integral without resorting to an analytic continuation to Euclidean space-time. The integrity of Minkowski space is thus preserved. 4 i.e. <H D ; D >=< D ;H D >. 5 Non-positive deniteness suggests it might becharge density. 6 Say, a real Grassmann variable, for the fermionic case 7

8 4.2 Gauge Invariance Finally, we indulge in some fanciful speculation: First, observe that transformations of the type D! e j D ( 2 R) (36) leave j D j 2 invariant. Gauging this global transformation (i.e. allowing to vary at dierent space-time points) augments the massless Dirac equation into the following expression: (@ jg ) D =: (37) Here, G is the associated gauge eld (the equation for the eld tensor F G has been excluded). The transformation (36) is not a Lorentz transformation, so what can it be? The similarities between the electromagnetic eld and the gauge eld G are rather striking. For example, in the source free case, the eld tensors are exactly equivalent. In the physical world, there are two forces which, classically, appear very similar; namely, the force between two stationary charges, and the force between two masses, each of which are described by aninverse square law. It is tempting, then, to view the U(1;D) gauge theory just presented as a simple gauge theory of gravity. We leave such speculations to the interested reader! 5 Concluding Remarks Our observations suggest that it is unnecessary (and potentially restrictive) to view complex-valued quantities as fundamental to a relativistic quantum theory. Quantum physics depends heavily on the concept of Hermitian operators, since it is precisely these operators which guarantee a real (i.e. measurable) spectrum of eigenvalues. Consequently, if we are seeking a quantum theory which is consistent with relativity an essential requirement for quantum gravity then it is appropriate that a unitary representation of the Lorentz group be considered. Such a representation can be obtained, but it requires the somewhat unorthodox step of embracing an unfamiliar number algebra. Hopefully, the eort of venturing beyond the familiar complex number system will be more than compensated by a new and fruitful perspective on the old quantum world. Acknowledgements I am indebted to many members of the Theoretical Physics and Mathematics Department (Oxford) for stimulating discussions on mathematical physics. This work was nancially supported by the Commonwealth Scholarship and Fellowship Plan (The British Council, U.K.), and is dedicated to I.W. References [1] A.Kwasniewski & R.Czech Reports on Mathematical Physics Vol 31 No. 3; June [2] Antonuccio F.A. Semi-Complex Analysis and Mathematical Physics gr-qc [3] Ryder, Lewis H 1992 Quantum Field Theory Cambridge University Press. [4] Aitchison & Hey 1989 Gauge Theories In Particle Physics 2nd Edition; Adam Hilger, Bristol. 8

9 [5] M.Nakahara 199 Geometry, Topology and Physics Adam Hilger, Bristol. [6] D.H. Sattinger & O.L. Weaver 1986 Lie Groups and Algebras with Applications to Physics, Geometry, and Mechanics. Springer-Verlag New york Inc. [7] L.I. Schi 1986 Quantum Mechanics 3rd Edition. McGraw-Hill Book Company, London. [8] R.Penrose & W.Rindler 1984 Spinors and Space-Time I Cambridge University Press. Cambridge. [9] M. Field 1982 Several Complex Variables and Complex Manifolds II. Cambridge University Press. Cambridge. [1] F. Mandl & G. Shaw 1984 Quantum Field Theory. John Wiley and Sons. New York. [11] Bailin & Love 1986 Introductionm to Gauge Field Theory. Graduate Student Series in Physics. Adam Hilger, Bristol. 9