The hidden geometric character of 7 0 0 2 relativistic quantum mechanics n a J 1 1 Jos´e B. Almeida 3 v Universidade do Minho, Physics Department 3 2 1 Braga, Portugal, email: [email protected] 6 0 6 0 / h Geometry canbe an unsuspected sourceof equations with physicalrele- p - vance, as everybody is aware since Einstein formulated the general theory t n a ofrelativity. Howevereffortstoextendasimilartype ofreasoningto other u q areasofphysics,namelyelectrodynamics,quantummechanicsandparticle : v i physics,usuallyhadverylimitedsuccess;particularlyinquantummechan- X r ics the standard formalism is such that any possible relation to geometry a is impossible to detect; other authors have previously trod the geometric path to quantum mechanics, some of that work being referred to in the tex. In this presentation we will follow an alternate route to show that quantum mechanics has indeed a strong geometric character. The paper makes use of geometric algebra, also known as Clifford al- gebra, in 5-dimensional spacetime. The choice of this space is given the 1 Jos´e B. Almeida The hidden geometric character of relativistic quantum mechanics character of first principle, justified solely by the consequences that can be derived from such choice and their consistency with experimental re- sults. Given a metric space of any dimension, one can define monogenic functions,thenaturalextensionofanalyticfunctionstohigherdimensions; suchfunctions have null vector derivative and have previouslybeen shown by other authors to play a decisive role in lower dimensional spaces. All monogenic functions have null Laplacian by consequence; in an hy- perbolic space this fact leads inevitably to a waveequationwith plane-like solutions. This is also true for 5-dimensional spacetime and we will ex- plore those solutions, establishing a parallel with the solutions of the free particle Dirac equation. For this purpose we will invoke the isomorphism between the complex algebra of 4 4 matrices, also known as Dirac’s ma- × trices. There is one problem with this isomorphism, because the solutions to Dirac’s equation are usually known as spinors (column matrices) that don’t belong to the 4 4 matrix algebra and as such are excluded from × the isomorphism. We willshowthat asolutionin termsofDirac spinorsis equivalent to a plane wave solution. Just as one finds in the standard formulation, monogenic functions can be naturally split into positive/negative energy together with left/right ones. This split is providedby geometric projectors and we will show that there is a second set of projectors providing an alternate 4-fold split. The possibleimplicationsofthisalternatesplitarenotyetfullyunderstoodand are presently the subject of profound research. 2 Jos´e B. Almeida The hidden geometric character of relativistic quantum mechanics 1 Introduction I have been advocating in recent papers that the majority of physics equations can be derived from an appropriately chosen geometry by exploration of the monogenic condition. Monogenic functions are not familiar to everybody but they are really the natural extension of analytic functions when one uses the formalism of geometric algebra[1, 2, 3, 4]; those functions zerothe vectorderivativedefined onthe algebraof the particular geometry under study. In [5] I showed how special relativity and the Dirac equation could be derived from the monogenic condition applied in the geometric algebra of 5-dimensional spacetime G . Anearlierpaper[6]provedthatthe sameconditioninthesamealgebrawassuf- 4,1 ficient to produce a symmetry group isomorphic to the standard model gauge group; unfortunately this paper is incorrect in the formulation of particle dynamics but the flaw was recently corrected;[7] the same work introduces electrodynamics and electro- magnetism in the monogenic formalism. Cosmologicalconsequences were drawn from the addition of an hyperspherical symmetry hypothesis with the consequent choice of hypersphericalcoordinates.[8]Summing upallthosecitedpapers,I wrotea longbook chapter;[9] the latest in the series is the derivation of energy states for the hydrogen atom from the monogenic condition.[10] The present paper uses the 5D monogenic condition in 5D spacetime as a postulate and explores its consequences for quantum mechanics, establishing the conditions for equivalence tofree particle Dirac’sequationbut goingbeyondthatandopening paths that may lead to particle physics. 3 Jos´e B. Almeida The hidden geometric character of relativistic quantum mechanics 2 Some geometric algebra Geometric algebra is not usually taught in university courses and its presence in the literature is scarce; good reference works are [1, 2, 3]. We will concentrate on the algebra of 5-dimensional spacetime because this will be our main working space; this algebra incorporatesas subalgebrasthose of the usual 3-dimensionalEuclideanspace, Euclidean4-spaceandMinkowskispacetime. Webeginwiththesimpler5-Dflatspace andprogresstoa5-Dspacetimeofgeneralcurvature(seeAppendixCformoredetails.) The geometric algebra G of the hyperbolic 5-dimensional space with signature 4,1 ( ++++)isgeneratedbythecoordinateframeoforthonormalbasisvectorsσ such α − that (σ )2 = 1, 0 − (σ )2 =1, (2.1) i σ σ =0, α=β. α β · 6 Note that the English characters i,j,k range from 1 to 4 while the Greek characters α,β,γ rangefrom 0 to 4. See Appendix A for the complete notation conventionused. Anytwobasisvectorscanbe multiplied,producingthenewentitycalledabivector. This bivector is the geometric product or quite simply the product, and it is distribu- tive. Similarly to the product oftwo basisvectors,the productof three differentbasis vectorsproducesatrivectorandsoforthuptothefivevector,becausefiveisthespatial dimension. We will simplify the notation for basis vector products using multiple indices, i.e. σ σ σ . The algebra is 32-dimensional and is spanned by the basis α β αβ ≡ 1 scalar, 1, • 4 Jos´e B. Almeida The hidden geometric character of relativistic quantum mechanics 5 vectors, σ , α • 10 bivectors (area), σ , αβ • 10 trivectors (volume), σ , αβγ • 5 tetravectors (4-volume), iσ , α • 1 pseudoscalar (5-volume), i σ . 01234 • ≡ Several elements of this basis square to unity: (σ )2 =(σ )2 =(σ )2 =(iσ )2 =1. (2.2) i 0i 0ij 0 The remaining basis elements square to 1: − (σ )2 =(σ )2 =(σ )2 =(iσ )2 =i2 = 1. (2.3) 0 ij ijk i − Note thatthe pseudoscalaricommuteswith allthe otherbasiselements while being a square root of 1; this makes it a very special element which can play the role of the − scalar imaginary in complex algebra. In 5-dimensional spacetime of general curvature, spanned by 5 coordinate frame vectors g , the indices follow the conventions set forth in Appendix A. We will also α assume this spacetime to be a metric space whose metric tensor is given by g =g g ; (2.4) αβ α β · the double index is used with g to denote the inner product of frame vectors and not their geometric product. The space signature is still ( ++++), which amounts to − saying that g < 0 and g > 0. The coordinate frame vectors can be expressed as a 00 ii linear combination of the orthonormed ones, for which reason we have g =nβ σ , (2.5) α α β 5 Jos´e B. Almeida The hidden geometric character of relativistic quantum mechanics where nβ is called the refractive index tensor or simply the refractive index; its 25 α elements can vary from point to point as a function of the coordinates.[11] In this work we will not consider spaces of general curvature but only the connection-free ones,whichcanbe designatedasbentspaces;inthosespaceswedefine the vectorand covariant derivatives (see appendix D). 3 Electromagnetism as gauge theory The simplest example of a gauge theory is electromagnetism, so we will start by analysing this in the scope of monogenic functions. The monogenic condition in flat space is given by[9] Ψ=0; (3.1) ∇ which has plane wave like solutions of the type Ψ=ψ eipαxα; (3.2) 0 where p = E is interpreted as energy, p = m as rest mass and p = σmp as 3- 0 4 m dimensional momentum. Because the second order equation 2Ψ = 0 must also be ∇ verified, we conclude easily that E2 p2 m2 =0; (3.3) − − that is Eσ0+p σm+mσ4 is a null vector. m A global symmetry of this equation is obtained with the transformation Ψ′ Ψeiβ, (3.4) 7→ where β is a constant. It is easily verified that if Ψ is monogenic so is Ψ′ and the symmetry is global because β is the same everywhere;the quantity exp(iβ) is a phase 6 Jos´e B. Almeida The hidden geometric character of relativistic quantum mechanics factor. If we allowβ to changewith spacetimecoordinates,β =β(xµ), the monogenic condition will no longer be verified by Ψ′ Ψ′ = Ψeiβ +iµ βΨeiβ. (3.5) ∇ ∇ ∇ Wecandefinealocalsymmetrybychangingthemonogenicconditionthroughreplace- mentofthevectorderivativebyacovariantderivativewhichcancelsouttheextraterm in Eq. (3.5); we do this as usual, by defining a vector field A=σµA and writing the µ covariant derivative as q D= µ + σ4+ A ∂ ; (3.6) 4 ∇ m (cid:16) (cid:17) where A transforms as µ 1 A′ A µ β. (3.7) 7→ − q ∇ In the two equations above q and m are charge and mass densities, respectively; in the non-dimensional units system q = 1 for an electron. Applying the extended − monogenic condition to Ψ′ we get D′Ψ′ = Ψeiβ+iqAΨeiβ =DΨeiβ. (3.8) ∇ So, obviously, if DΨ is null, that is if Ψ is monogenic in an extended sense, so is Ψ′. Now, the covariantderivative in Eq.(3.6) can be associatedwith the reciprocalframe q gµ =σµ, g4 =σ4+ A σµ, (3.9) µ m which was introduced in Refs. [7, 9] to derive electrodynamics and electromagnetism fromthemonogeniccondition. Weseethenthatelectromagneticgaugeinvarianceisa consequence of a non-orthonormed g4 frame vector and can be accommodated in the refractive index formalism introduced above. 7 Jos´e B. Almeida The hidden geometric character of relativistic quantum mechanics 4 The Dirac equation In this section we examine the monogenic condition (3.1) and the solution (3.2) to establish the conditions under which they become equivalent to free particle Dirac’s equation; the section ends with an indication of the procedure for the inclusion of an EM field. We will accept without explanation that the solution Ψ has harmonic dependence on x4 with a frequency equal to a particle’s rest mass and write Ψ=ψeimx4. (4.1) At the present moment we cannot offer a plausible explanation for the existence of eigenvalues for the rest mass, corresponding to the various elementary particles; this is still a mystery with which we must live if we want to proceed. We note, however, that no existing theory offers a satisfactory explanation for the observed elementary particles’ masses and so we are no worse than all other theories in that respect. When the monogenic condition (3.1) is applied to Ψ with the assumption above we get (σ0∂ +σm∂ +iσ4m)ψ =0; (4.2) t m multiplying by iσ0 on the left (i∂ +iσm0∂ σ40m)ψ =0. (4.3) t m − Geometric algebra G is isomorphic to the complex algebra of 4 4 matrices [12], 4,1 ∗ which means we can representby matrix equations all the geometric equations in this paper. By resorting to matrix equations we will loose the geometric content inherent to geometric algebra; nevertheless such step is crucial if we want to understand the relationship with Dirac’s equation because the latter uses matrix formalism. In order 8 Jos´e B. Almeida The hidden geometric character of relativistic quantum mechanics Table 1: Dirac-Pauli matrices. 0 0 0 1 0 0 0 i −     0 0 1 0 0 0 i 0 α1 = , α2 = ,     0 1 0 0 0 i 0 0        −      1 0 0 0 i 0 0 0         0 0 1 0 1 0 0 0     0 0 0 1 0 1 0 0 α3 = − , β = .     1 0 0 0  0 0 1 0         −      0 1 0 0  0 0 0 1  −   −      tomakethetransitionbetweenthetwoequationsweneedamatrixrepresentationfor geometric algebra elements; although there are many possible choices we will adopt the Dirac-Pauli representation, represented in Table 1, because it is commonly found in the literature. We can now associate geometric algebra elements to their matrix counterparts σm0 αm, σ40 β. (4.4) ≡ ≡ The different basis vectors of geometric algebra can then be given matrix equivalents 9 Jos´e B. Almeida The hidden geometric character of relativistic quantum mechanics Table 2: Matrix representation of basis vectors 0 0 1 0 0 1 0 0 0 i 0 0 −       0 0 0 1 1 0 0 0 i 0 0 0 σ0 , σ1 , σ2 ,       ≡ 1 0 0 0 ≡0 0 0 1 ≡0 0 0 i       −   −           0 1 0 0 0 0 1 0  0 0 i 0  −   −   −        1 0 0 0 0 0 1 0 −     0 1 0 0 0 0 0 1 σ3 − , σ4 − .     ≡0 0 1 0 ≡ 1 0 0 0       −  −      0 0 0 1  0 1 0 0     −      by σ0 iα1α2α3β, ≡ σ1 iα2α3β, ≡ σ2 iα1α3β, (4.5) ≡ σ3 iα1α2β, ≡− σ4 iα1α2α3; ≡ resulting in the matrices of Table 2 The monogenic condition (3.1) can then be written in the alternative form (i∂ +iαm∂ βm)ψ =0; (4.6) t m − this can be read either as a geometric or matrix equation. In the latter case one recognizesthatit has become Dirac’s ownequationin the absence offields. There are 10