Found Phys (2008) 38: 610–647 DOI 10.1007/s10701-008-9223-3
The Foundations of Quantum Mechanics and the Evolution of the Cartan-Kähler Calculus Jose G. Vargas
Received: 20 July 2007 / Accepted: 5 May 2008 / Published online: 16 May 2008 © Springer Science+Business Media, LLC 2008
Abstract In 1960–1962, E. Kähler enriched É. Cartan’s exterior calculus, making it suitable for quantum mechanics (QM) and not only classical physics. His “KählerDirac” (KD) equation reproduces the fine structure of the hydrogen atom. Its positron solutions correspond to the same sign of the energy as electrons. The Cartan-Kähler view of some basic concepts of differential geometry is presented, as it explains why the components of Kähler’s tensor-valued differential forms have three series of indices. We demonstrate the power of his calculus by developing for the electron’s and positron’s large components their standard Hamiltonian beyond the Pauli approximation, but without resort to Foldy-Wouthuysen transformations or ad hoc alternatives (positrons are not identified with small components in Kähler’s work). The emergence of negative energies for positrons in the Dirac theory is interpreted from the perspective of the KD equation. Hamiltonians in closed form (i.e. exact through a finite number of terms) are obtained for both large and small components when the potential is time-independent. A new but as yet modest new interpretation of QM starts to emerge from that calculus’ peculiarities, which are present even when the input differential form in the Kähler equation is scalar-valued. Examples are the presence of an extra spin term, the greater number of components of “wave functions” and the non-association of small components with antiparticles. Contact with geometry is made through a Kähler type equation pertaining to Clifford-valued differential forms. Keywords Foundations of quantum mechanics · Cartan-Kähler calculus · Foldy-Wouthuysen · Positron’s Hamiltonian
J.G. Vargas () PST Associates, 48 Hamptonwood Way, Columbia, SC 29209, USA e-mail:
[email protected]
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1 Introduction In the mid to late 1920’s, a few very gifted scientists developed the present formalism of quantum mechanics (QM), and of quantum electrodynamics (QED). The merits of such formalism not withstanding, it has to be seen as an ad hoc response to the needs of the physics of the time. Eventually the calculus of differential forms evolved to the point where it could address the needs of QM. In this paper, we shall show that such a calculus constitutes a much better way to compute in QM, specifically in relativistic QM. In the last three sections, we shall also show the new perspective that it brings into the foundations of QM, even if the computations produce so far the same results as with the Dirac formalism. An important highlight in the evolution of the calculus is constituted by Cartan’s introduction in a paper on differential equations of his exterior calculus of differential forms [1]. It is the modern language of differential geometry, differential topology and other branches of mathematics. It is based on exterior algebra. For decades, this calculus remained basically ignored except in Cartan and Kähler’s work. The latter used it in 1934 to generalize the former’s theory of exterior differential systems [2]. In the early 1960’s, Kähler generalized the exterior calculus itself, by endowing differential forms with richer algebraic structure [3–5]. Kähler’s sophisticated calculus revolves around a basic equation that parallels the equation df/dx = gf . It is called the Kähler-Dirac or (simply) Kähler equation, (12), to which he still referred as Dirac’s. Among the differences between the Dirac and Kähler equations (both give the same fine structure for the hydrogen atom [4, 5]), there is the replacement of spinors with inhomogeneous differential forms with complex coefficients, differential forms which have 16 independent complex components. This is a very significant feature, whose consequences for QM remain to be explored. Kähler’s work, written in German and not yet translated, constitutes a formidable piece of mathematics. His theory is relativistic ab initio, but one does not need to know relativity to be able to use it. In fact, dealing with the issues that are normally associated with relativistic QM (exception made of solving the hydrogen atom) requires just the simple version of this calculus pertaining to the use of Cartesian coordinates and scalar-valued forms. Being equation-based, a prior theory of physical observables is not needed, at least for the purposes that occupied Kähler and that will occupy us in this paper. Absent are also the gamma matrices. An understanding of the Kähler calculus encounters the further barrier that one should be familiar with Cartan’s methods, which Kähler largely follows. A related complication is that he does not explain why components in his calculus have one series of superscripts and two series of subscripts. It is, however, clear that the widely held view that the calculus of differential forms pertains to antisymmetric covariant tensors clashes with the realities of this calculus, as it would appear that there are two types of antisymmetric covariant tensors. For this reason, we shall present in Sect. 2 the Cartan-Kähler approach to vector fields and tensor-valued differential forms, as well as to exterior, interior, covariant and Lie derivatives. Fortunately, quantities with just one series of indices are sufficient to deal with problems such as the fine structure of the hydrogen atom [4, 5] and everything that we shall do in this paper.
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In Sect. 3, we summarize several outstanding features of his calculus and his equation, features which he himself discussed. Some knowledge of Clifford algebra is assumed. In Sect. 4, we briefly present further developments dealing with what seems to be ad hoc or unexplained in his calculus. In the same section, we also raise the issue of two “Kähler-Dirac” equations, respectively from 1960 and 1962, for when the input differential forms are tensor-valued. Except for Sect. 11, the whole paper will be confined to scalar-valued input, whose components, like those of the output (or wave) differential form, have just one series of indices. Kähler did not study any particular problem where the input was not scalar-valued. In Sect. 5, we discuss rotations, angular momentum, spinors and plane waves, all of it in the context of the two important concepts of Killing symmetries and constant differentials. Readers thoroughly familiar with this calculus could just go directly to Sects. 6 to 8. They might solve themselves the “graduate problems” that we proceed to formulate. That solving shows the power to do with the Kähler calculus what the Dirac formalism can only do with hole theory plus Foldy-Wouthuysen transformations (or similar paraphernalia), if at all. Let ∨ stand for Clifford product in the Kähler algebra of scalar-valued differential forms, algebra whose elements can be written as sums of differential forms of different grades, thus called inhomogeneous. Those sums have odd and even grade parts. Let − be the idempotent defined in (29). Multiplied on the right of the wave differential form, it annuls the positron contribution to it. In other words, it annuls what in Kähler’s theory plays the role of the negative energy components of the Dirac spinors, except that the energies in the positron sector are now of the same sign as in the electron’s (see Sect. 3(f)). As also explained in Sect. 3(f), the electron part can be written so that dt appears only in − , which is placed as the last factor in the differential form that solves the Kähler equation with minimal electromagnetic (EM) coupling. Let ∂j denote partial derivative with respect to x j . Here are the problems that we solve (all except the last one concern arbitrary EM fields). Problem 1: At low energy, write the wave differential form u of an electron in a 2 weak EM field as e−imc t/ R(x i , t, dx i ) ∨ − , where m is the mass of the electron, where Latin indices run from 1 to 3, and where the differential form R depends on t, but not on dt. Obtain from the KD equation with minimal EM coupling, (59), the system of equations satisfied by the even (ϕ) and odd (χ ) parts of R. i Problem 2: Define P ≡ dx j ∨ (−i∂j − eAj ) and χ1 ≡ − 2m P ∨ ϕ. Replacing χ with χ1 in the system of problem 1, obtain the Pauli equation for ϕ. Problem 3: Show that, in the next order of approximation for χ that the system itself suggests, one obtains in essence the same development of the Hamiltonian for ϕ as, say, in Bjorken and Drell [6], but without the need for Foldy-Wouthuysen transformations. 2 Problem 4: With positrons given in the form e−imc t/ S(x i , t, dx i ) ∨ + , repeat (or just see the pattern emerge) the same tasks as in problems 1 to 3, again for large components, which constitute now the odd part χ of S. Problem 5: Transform the coupled system of equations resulting in problem 1, or in problem 4, so that the roles of ϕ and χ are exchanged. Relate to the negative
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energy solutions of the Dirac theory the term that, under that transformation, jumps from one equation to the other. Problem 6: Based on the behavior of the lowest order approximations to χ in problems 1 and 2, develop the iterative procedure that suggests itself for obtaining χ , and thus the Hamiltonian for ϕ, in ever increasing orders. Problem 7: The wave form u of an electron in a constant EM field may also be 2 written as u = e−iEc t/ (x i , t, dx i ) ∨ − , where E is the energy of the system and where all the dt dependence again is inside − . Equate the expressions for u respectively in terms of (R, m) and (, E) so as to obtain simple expressions for ϕ,t (also χ,t ) in terms of ϕ (respectively χ ). Use those expressions to readily obtain Hamiltonians for ϕ and χ that contain just a few terms but are exact (i.e. closed form expansion of the Hamiltonian). A point about terminology follows. Algebraists make subtle differences between the concepts that go by the terms of exterior and Grassmann algebras, and even more with the use of the terms Grassmann algebra, Clifford algebra and Clifford-Atiyah algebra. Whereas exterior algebra is based on just the exterior product, Grassmann algebra is sometimes said to contain the additional structure conferred by an inner product induced by a quadratic form [7]. In the same reference, Clifford algebra is said to contain only the Clifford product (example, the algebra of Dirac gamma matrices) [7]. The Kähler-Atiyah algebra contains both the Grassmann and Clifford algebras as substructures [7]. However, the Clifford product of two vectors can be decomposed into their symmetric and antisymmetric parts, identifiable with their exterior and interior products (similarly for the products of a vector and a multivector, but the general case is more complicated). Hence, we adopt the practical perspective of using the term Clifford algebra for the Kähler-Atiyah algebra, and the term Grassmann algebra for exterior algebra. The term Kähler algebra will be reserved specifically for the so understood Clifford algebra (i.e. Kähler-Atiyah algebra) of inhomogeneous cochains, the homogeneous r-cochains being functions of r-surfaces. As we shall explain later in the paper, we shall use the term differential forms for these cochains, in accordance with Kähler, since they, rather than the antisymmetric multilinear functions of vectors, are of the essence of his calculus [8]. In Sect. 9, we present the more Einsteinian perspective of QM that appears to emerge from the Kähler equation. Sections 10 and 11 deal respectively with what this author considers to be the “bottom up” and “top down” research priorities in this field. As a reviewer clearly put it, the Kähler-Cartan calculus will have a real importance in modern theoretical and mathematical physics if the author and/ or other authors will elaborate in further works new results of relevance, or else its importance will be simply historical.
2 The Cartan-Kähler View of Basic Concepts of Geometry and the Calculus In the comprehensive treatment by Choquet-Bruhat et al. [9], tangent vectors are defined (a) as sets of quantities that transform in a particular way, (b) as linear operators on spaces of functions (thus playing active roles) and (c) as equivalence classes of curves (passive vectors, not acting on anything). Defining concepts by their transformation properties is clearly inadequate. For instance, a 1-cochain, i.e. a function
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of curves, and a field of linear functions of vectors transform in the same way. Similarly, a tangent vector field referred to the reciprocal basis ei (defined by ei = g ij ej ) has components which transform like 1-cochains and like fields of linear functions of vectors (in this section, i goes from 1 to the dimension n of the space). Based just on transformation properties, there would be no point in having two series of subscripts in the Kähler calculus. The vectors defined in (b) are operators consisting of linear combinations of the partial derivatives with respect to the coordinates. Cartan refers to them as infinitesimal transformations [10], not as tangent vectors. Specifically, he does not use the term vector field when he lets an infinitesimal transformation act on a differential rform to obtain another differential r-form (i.e. its Lie derivative). Nor does he use it when he generates a differential (r − 1)-form by the action of a differential r-form on an infinitesimal operator, nowadays called the evaluation of a differential form on a vector field [10]. Cartan and Kähler’s tangent vectors do not play the active role that the infinitesimal operators play. We proceed to explain the approaches to differentiation by Cartan and Kähler. The former uses exterior differentiation. Kähler uses more general differentiations, containing interior and exterior parts. The latter part constitutes Cartan’s exterior derivative, or simply Cartan’s derivative. It gives the exterior derivative when acting on scalar-valued differential forms, and the covariant derivative when acting on tensor-valued differential 0-forms, i.e. on tensor fields (his rather casual approach to differentiation has been formalized by Flanders [11]). Those authors use the term exterior even when d acts on vector-valued forms, dd then not yielding zero in general (see (3)). Kähler’s combined exterior-interior differentiation (which he called interior differentiation) does not satisfy the Leibniz rule, not even when applied to scalar-valued differential forms. Following Kähler, we still retain the use of the term differentiation. We return to vector fields. They are passive in treatments of the theory of curves and surfaces in Euclidean space, like in books by Struik [12] and H. Cartan [13], or in treatments of the tensor calculus, like in the authoritative book by Lichnerowicz [14] (in the following, as was the case in previous paragraphs, Cartan will mean É. Cartan). Of course, one can do flat space geometry with active vector fields, but it is not intuitive and runs counter to what one learned in, say, courses on the multivariable calculus. The key point here is that the passive concept of a tangent vector as an equivalence class of curves at a point bodes well with the simple concept of vector field that is standard in the study of Euclidean space. In both cases we are dealing with passive vector fields, however defined. Exterior differentiation of v in the extended sense of Cartan, Kähler and Flanders yields .j
dv = d(v i ei ) = dv i ei + v i dei = dv i ei + v i ωi ej .
(1)
Differentiating again, one gets .j
.j
.j
ddv = (0 − dv i ∧ ωi ej ) + (dv i ∧ ωi ej + v i dωi.k ek − v i ωi ∧ ωj.k ek ).
(2)
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We have used parentheses to make clear the origin of each of the five terms on the j right of (2), respectively from dv i ei and v i ωi ej . We further have .j
.j
ddv = v i (dωi − ωi ∧ ωj.k )ek = v i Ωi.k ek .
(3)
In general, the Ωi.k are meant to refer to given bivector-valued differential forms from which one tries to find the connection through integration. As explained by Lichnerowicz [14] (Sects. 1.10 and 1.11), the v i play the role of a basis of linear forms (a field thereof in this case). Hence one has to distinguish .k v i e ωj ∧ ωl that between the basis field v i and the basis ωm in the expression Ri.j k l .k .k j l results from replacing Ωi with Ri.j l ω ∧ ω on the right hand side of (3). Consider next the concept itself of differential forms. Modernly, these are defined as fields of antisymmetric multilinear functions of vectors, though, in the same breath, some authors say that they are integrands. Rudin [15], however, defines differential r-forms as the integrands themselves, i.e. as functions of r-surfaces, whose evaluation is their integration. Only the last definition is consistent with the use made in the Kähler calculus of the standard symbols of the exterior calculus. In both calculi, the same differentiation operator yields different derivatives depending on the mathematical object it acts upon: on the one hand, multilinear functions of vectors, whether antisymmetric or not, and, on the other hand, functions of r-surfaces. The general formula that Kähler gives for derivatives speaks of the nature of the object to which each series of subscripts refers. The exterior part d of his derivative is the usual covariant one if the second series of subscripts is empty. But d is the exterior derivative in the absence of superscripts and first series of subscripts. It is clear that, since the action of d (in the sense of Cartan, Kähler and Flanders) on tangent tensors depends on connection, its action on multilinear functions of tangent vectors, whether antisymmetric or not, must also depend on connection. On the other hand, functions of r-surfaces (and grade-inhomogeneous combinations thereof) live on the base manifold, not on bundles. The action of d on them does not depend on connection. It follows that what Kähler, Cartan, Rudin and this author call (and write with the symbol of) differential r-forms are functions of r-surfaces, not antisymmetric r-linear functions of vectors. Consider finally Lie derivatives. In a chapter titled “The differential systems that admit an infinitesimal transformation” [10], Cartan extends the concept of such transformations from the ring of functions to the ring of differential forms. The extension is a matter of straightforward computations once the equation Xdx i = dXx i has been reached, X being the infinitesimal transformation. Though straightforward, the computation of the action of infinitesimal transformations on differential forms is cumbersome. Cartan avoids it and produces a theorem relating the action of infinitesimal operators on forms to the action of forms on infinitesimal operators. Since the latter action is very easy to compute, the theorem can thus be used to easily obtain the first action, i.e. what is nowadays known as the Lie derivative of the form. Hence, he produces after all a formula which can be used to easily compute the Lie derivative of differential forms, even if it is presented as a theorem with a different purpose. When referring to the work by Cartan just mentioned, Kähler uses the term Lie derivative instead of infinitesimal transformation. He reaches a formula for comput-
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ing it [3]. It does not take quite the same form as the equation that states the aforementioned Cartan theorem. His derivation is somewhat more cumbersome but less recondite. With α i defined through X ≡ α i ∂/∂x i , he constructs the differential system dx i = α i (x 1 , . . . , x n ), dλ
i = 1, . . . , n.
(4)
Kähler then defines a new coordinate system y. Its y n coordinate is λ, and the y i (i = 1, . . . , n − 1) are n − 1 first integrals, independent among themselves and of the specific integration constant that is additive to λ and constitutes the nth independent first integral. The pull-back to the y coordinate system of a differential p-form 1 ai ...i dx i1 ∧ · · · ∧ dx ip p! 1 p
(5)
1 ∂x i1 ∂x ip ai1 ...ip k · · · k dy k1 ∧ · · · ∧ dy kp . p! ∂y 1 ∂y p
(6)
u= is u=
Kähler shows that acting on (6) with
∂ ∂y n
one gets
∂u 1 = (Xai1 ...ip )dx i1 ∧ · · · ∧ dx ip + d(α i ) ∧ ei u, n ∂y p!
(7)
where Xai1 ...ip is, of course, α j ∂j ai1 ...ip . Formula (7) is a verbatim copy of the original, except for the parenthesis around Xai1 ...ip . It should be written (also (6)) with a pull-back symbol on the left hand side. The operator ei is a contraction: one places the factor dx i at the front of each string of products of differential 1-forms, with a change of sign if applicable, and eliminates that factor. This action yields zero on terms where dx i is not present. In 1962, Kähler defined the Lie derivative of a tensor-valued differential form. He did so seeking (verbatim) that k ...k
(σ Xu)i11...iλμ =
∂ k ...k (σ u)i11...iλμ , ∂y n
(8)
where σ is the pull-back operator to the y coordinate system (symbol which he uses instead of σ ∗ ) and where the k and i indices are valuedness indices. Each of k ...k the (σ u)i11...iλμ for different values of the exhibited indices is a differential form; hence a third series of indices, the differential form indices, is implicit on both sides of (8). Kähler does not resort to any commutator of his operator X with the tangent and cotangent (valuedness) fields. He proves that, in terms of arbitrary coordinate sys-
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tems, (8) implies: k ...k
(Xu)i11...iλμ = α i
∂ k1 ...kμ k ...k ui1 ...iλ + d(α i ) ∧ ei ui11...iλμ i ∂x
∂α k1 r...kμ ∂α kμ k1 ...r ui1 ...iλ − · · · − u r ∂x ∂x r i1 ...iλ ∂α r k1 ...kμ ∂α r k1 ...kμ + i ur...i + · · · + u . λ ∂x 1 ∂x iλ i1 ...r −
(9)
Commutation relations still emerge in any case from (9), and thus from (8) in the last instance. Indeed, formula (9) becomes Xuk = α i
k ∂uk i ∂α − u ∂x i ∂x i
(10)
for a vector field uk . Formula (9) is thus consistent with the modern concept of Lie derivative of a vector field as the commutator, LX Y = [X, Y ] ≡ XY − Y X.
(11)
In spite of the correspondence between (8) and (11), the right hand side of (8) may give the impression that something is wrong with it, given the absence of the commutator. This is not a real absence; it simply happens that the −Y X term is zero in the y coordinate system, where the components of X are (0, 0, . . . , 1). Finally, as previously reported, Cartan did not define Lie derivatives (i.e. infinitesimal operators) of vector fields. Kähler did, but one wonders what did he want them for, since he did not use them. It is worth recalling in connection with (9) the important point made before to the effect that the symbols said to represent differential forms do represent cochains in Kähler’s work. In this regard it is instructive to consider Slebodzinski’s treatment of Lie differentiation [16]. Let v be a vector field. He obtains Xv using that the contraction of u with the differential 1-form β is a scalar. He uses for that the Lie derivative of β and the Leibniz rule. However, in first obtaining the formula for the Lie derivative of differential forms, Slebodzinski treats them as integrands, i.e. as functions of r-surfaces, whose d derivative does not depend on connection. Thus, he actually contracts the vector field with a cochain, not with a linear function of vector fields. Lie derivatives often enter the definition of differential geometers for affine curvature and torsion, definitions from which they obtain the equations of structure. Those equations can, however, be derived in a much simpler yet rigorous way and fully in the spirit of Cartan [17]. It is worth observing also that equality of two tangent vectors at two different points of a differentiable manifold (and, therefore, any geometrically significant derivative of vector fields and their duals) is determined by the affine connection, not by Lie derivatives. Sure enough one can have both concepts of derivatives (exterior and Lie) of vector fields. One should, however, remain aware of the limitations of having a concept of Lie derivative of differential forms when these are tensor-valued; the increments in the numerators of the limiting process involved
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in obtaining the right hand side of (8) lacks meaning because one needs a connection to compare the tangent spaces at the points x and x + x. No connection enters the Lie derivative equations, (8)–(11). Symmetries do not remove the necessity of using a connection to compare those tangent spaces. Clearly, the preceding comments do not apply to Lie derivatives in the theory of dynamical systems in flat spacetimes.
3 Quantum Mechanical Issues Raised by Kähler’s Own Work The combination of Dirac’s authoritative formalization of the principles of QM [18] and the Copenhagen interpretation did not constitute a solid enough foundation to preempt what Mead [19] (p. 5) refers to as the “muddle and fuss over theory” in the decades that followed. With regards to the Dirac equation in particular, Thaller [20] had this to say in the preface of his book on this subject: “Perhaps one reason that there are comparatively few books on the Dirac equation is the lack of an unambiguous quantum mechanical interpretation. Dirac’s electron theory seems to remain a theory with no clearly defined range of validity, with peculiarities at its limits which are not completely understood.” All that is not surprising since the development of those principles and interpretation took place to accommodate the new experimental evidence through the adoption of new versions of concepts inherited from a tradition where matter is particles to the exclusion of any wave concept. That course of action may be unavoidable in any useful quantum physics. However, more refined perspectives on their use may arise if those concepts are not introduced like with forceps, as happened in the 1920’s, in order to reach wave and evolution equations for observables. In contrast, the Kähler equation emerges naturally in the growth of the calculus of differential forms. Dirac’s theory, timely and magnificent, was nonetheless an ad hoc response to the needs of the physics. Nowadays, it may not be the best response. Kähler did let the mathematics speak on the subject of particles and antiparticles, and proposed an extended expression for angular momentum ([5], Sect. 27), which he did not, however, interpret physically. In Sect. 9, building upon results in Sect. 5, we make a modest incursion on an interpretation. We shall, however, keep interpretations to a minimum until the mathematics speaks more clearly through future computations. We proceed to report on highlights of his calculus that need emphasis and diffusion. Henceforth and because of the physical applications, we shall use Latin indices for 3-space, and Greek indices otherwise. (a) The Kähler equation is ∂u = a ∨ u,
(12)
where ∨ stands for Clifford product of the differential form factors of his tensorvalued differential forms, and where a and u are the input and output differential forms. For minimal EM coupling a is m+eA, up to universal constants. In connection with this equation it is important to notice that Kähler uses deficient notation for the product of his tensor-valued differential forms; he makes explicit the Clifford product
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symbol in the differential form algebra, but not the tensor product symbol in the valuedness algebra. For more on this see Sect. 4(c) in this paper. ∂ designates Kähler’s general derivative operator, except in ∂/∂x k and in ∂k , where it retains the meaning of partial derivative. It constitutes what we shall refer to as the sum of the exterior and interior derivatives (or exterior and interior covariant derivatives when ∂ acts on tensor-valued differential forms): ∂u = du + δu,
(13)
du = dx μ ∧ dμ u,
(14)
δu = dx μ · dμ u,
(15)
where
and where “·” stands for interior or dot product. Kähler only deals with the LeviCivita connection. He proves that his δu becomes ∗ d ∗ or -∗ d ∗ , the sign depending on signature and dimensionality of the space. This result may or may not be valid for other connections, depending on some freedom existing on how one defines dμ u in general. For present purposes, we do not need the details of what goes in the general case into the making of the operator dμ , and which Kähler calls the covariant derivative. In later sections, we shall deal with the simple expression that dμ takes in the specific case of rectilinear coordinates and scalar-valued differential forms. Suffice to say that it allows one to view the exterior derivative d and (in the case of the Levi-Civita connection) the coderivative δ in the new light of (14) and (15). The operators d and dμ satisfy the Leibniz rule, but ∂ and δ do not, even if acting on scalar-valued differential forms. Equations (13)–(15) support the use of the term derivative in all those cases, the modern use of the term not withstanding. It is to be noted that, if the exterior-interior calculus of differential forms had been formulated before the advent of QM, the natural order of differential equations to be solved might have been as follows: dμ u = 0 (which defines constant differentials), ∂u = 0 (strict harmonic differentials), ∂∂u = 0 (harmonic differentials), ∂u = a and ∂u = a ∨ u (of the type ∂u = a is the Maxwell system, where a is the differential 1-form dual to the EM current 3-form and where u is the EM differential 2-form F ). In other words, the basic equation of QM would have emerged in the calculus without input from the physics had mathematical knowledge been far ahead of where it has been. Of particular interest because of its novelty and properties is the just mentioned concept of constant differentials. They have the important property that, if u is a solution of a Kähler equation and c is a constant differential, u ∨ c also is a solution of the same Kähler equation, i.e. ∂(u ∨ c) = a ∨ (u ∨ c).
(16)
This concept will prove its worth when relating the solutions of the Kähler equation (with their 32 real components) to the spinors which solve Dirac equations.
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Also of interest is the dependence of the Laplacian on the curvature, if, as is natural, one defines the former as ∂∂u. Indeed, let eμ be the same operator as in (7), and define eμ as g μν eν . The operator ∂ satisfies [5] ∂∂u = g μν dμ dν u + Rμν dx μ ∨ eν u − Ωμν ∨ eμ eν u,
(17)
regardless of whether u is scalar-valued or not. Rμν stands for the Ricci contraction of the curvature tensor. To summarize, Kähler’s common mathematical formalism for QM and general relativity is constituted by his generalization of Cartan’s calculus of differential forms. Even if we were to constrain the Kähler calculus to scalar-valued differential forms, it would still constitute a generalization of the Dirac calculus with gamma matrices thanks to the additional possibilities opened by the fact that it depends on 32 rather than 8 independent real functions. (b) The conservation law of a bilinear expression on differential forms is built into the Kähler calculus through a generalized Green’s formula. A property of the Kähler equation then gives rise to a conserved quantity quadratic in the wave differential form, as we now explain. Let r and s be any two differential forms. Kähler defined a n-form r, s and a (n − 1)-form r, s1 by r, s ≡ (ζ s ∨ r)0 z,
r, s1 ≡ (ζ s ∨ dx μ ∨ r)0 eμ z,
(18)
where ( _ )0 denotes the 0-form part of the contents of the parenthesis, where ζ denotes reversion of the order of all the 1-form factors in differential forms, and where z is the unit differential n-form. It is worth pointing out that r, s =
n
rm , sm ,
(19)
m=0
and rm , sm =
(eμ1 · · · eμm rm )(eμ1 · · · eμm sm ) z,
(20)
i1
