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Mathematical propositions referring only to concrete objects in this sense Hilbert called real , concrete , or contentual propositions, and all other mathematical propositions he distinguished as possessing an ideal , or abstract character. Hilbert may be seen to have followed Kant in attempting to ground mathematics on the apprehension of spatiotemporal configurations; but Hilbert restricted these configurations to concrete signs such as inscriptions on paper.
Hilbert regarded consistency as the touchstone of existence, and so for him the important thing was the fact that no inconsistencies can arise within the realm of concrete signs, since correct descriptions of concrete objects are always mutually compatible. In particular, within the realm of concrete signs, actual infinity cannot generate inconsistencies since, again along with Kant, he held that this concept cannot correspond to any concrete object. Whether that program could be successfully carried out was, of course, still an open question.
But still:. Hence his memorable characterization:. But it must have been painful for him to concede the analogous claim in the case of mathematics. Here Weyl aims at a mathematical and philosophical elucidation of the problem of space and time in general. In the preface to the great classical work of , the fifth German edition, after mentioning the importance of mathematics to his work, Weyl says:.
Using the Einstein summation convention [ 34 ] , equation 1 can simply be written as. Riemann assumed the validity of the Pythagorean metric only in the infinitely small. Riemannian geometry is essentially a geometry of infinitely near points and conforms to the requirement that all laws are to be formulated as field laws. Field laws are close-action-laws which relate the field magnitudes only to infinitesimally neighbouring points in space.
The field magnitudes consist of partial derivatives of position functions at some point, and this requires the knowledge of the behavior of the position functions only with respect to the neighbourhood of that point. To construct a field law, only the behavior of the world in the infinitesimally small is required. Riemann had already suggested that analogous to the electromagnetic field, the metric field reciprocally interacts with matter. Einstein came to this idea of reciprocity between matter and field independently of Riemann, and in the context of his theory of general relativity, applied this principle of reciprocity to four dimensional spacetime.
After Einstein applied Riemannian geometry to his theory of general relativity, Riemannian geometry became the focus of intense research. In particular, G. Ricci and T. The decisive step in this development, however, was T. Weyl simply referred to the latter as affine connection. This definition merely says that a manifold is affinely connected if it admits the process of infinitesimal parallel displacement of a vector. The definition says that at any arbitrary point of the manifold there exists a geodesic coordinate system such that the components of any vector at that point are not altered by an infinitesimal parallel displacement with respect to it.
According to Weyl b, , it characterizes the nature of an affine connection on the manifold. A manifold which is an affine manifold is homogeneous in this sense. Moreover, manifolds do not exist whose affine structure is of a different nature. It is important to understand that there is no intrinsic notion of infinitesimal parallel displacement on a differentiable manifold.
It was Weyl b who emphasized and developed the metric-independent construction of the symmetric linear connection and who pointed out the rationale for doing so. In both the non-relativistic and relativistic contexts, it is the symmetric linear connection, and not the metric, which plays the essential role in the formulation of all physical laws that are expressed in terms of differential equations.
It is the symmetric linear connection that relates the state of a system at a spacetime point to the states at neighboring spacetime events and enters into the differentials of the corresponding magnitudes. In both Newtonian physics and the theory of general relativity, all dynamical laws presuppose the projective and affine structure and hence the Law of Inertia. In fact, the whole of tensor analysis with its covariant derivatives is based on the affine concept of infinitesimal parallel displacement and not on the metric.
In particular, it led to. The interest in projective geometry is in the paths , that is, in the continuous set of points of the image set of curves , rather than in the possible parameter descriptions of curves. A curve has one degree of freedom; it depends on one parameter, and its image set or path is a one-dimensional continuous set of points of the manifold. Consequently, two curves are mathematically considered to be different curves if they are given by different maps different parameter descriptions , even if their image set, that is, their path, is the same.
A path is therefore sometimes defined as an equivalence class of curves under arbitrary parameter transformations. Hence, projective geometry may be defined as an equivalence class of affine geometries. A geodesic curve in flat space is a straight line. Its tangent at one point is parallel to the tangent at previous or subsequent points. A straight line in Euclidean space is the only curve that parallel-transports its own tangent vector. This notion of parallel transport of the tangent vector also characterizes geodesic curves in curved space.
Weyl took the latter approach. According to Weyl, the infinitesimal process of parallel displacements of vectors contains, as a special case, the infinitesimal displacement of a direction into its own direction. Such an infinitesimal autoparallelism of directions is characteristic of the projective structure of an affinely connected manifold. This characterization of a geodesic curve constitutes an abstraction from affine geometry. Through this abstraction, a geodesic curve is definable exclusively in terms of autoparallelism of tangent directions , and not tangent vectors.
Roughly speaking, an affine geometry is essentially a projective geometry with the notion of distance defined along the curves. By eliminating all possible notions of distance along curves, or equivalently, all the parameter descriptions of the curves, one abstracts the projective geometry form affine geometry. Weyl presented the details of his approach to projective geometry, which uses the notion of autoparallelism of direction , in a set of lectures delivered in Barcelona and Madrid in the spring of Weyl a ; see also Weyl c.
In physical spacetime the projective structure has an immediate intuitive significance according to Weyl. It is an indubitable fact, according to Weyl a, 13 , that a body which is let free in a certain spacetime direction time-like direction carries out a uniquely determined natural motion from which it can only be diverted through an external force.
If external forces exert themselves on a body, then a motion results which is determined through the conflict between the tendency of persistence due to the guiding field and the force. The tendency of persistence of the guiding field is a type of constraining guidance, that the inertial-gravitational field exerts on every body. Weyl b, says:. Shortly after the completion of the general theory of relativity in , Einstein, Weyl, and others began to work on a unified field theory. It was natural to assume at that time [ 44 ] that this task would only involve the unification of gravity and electromagnetism.
In the general theory of relativity the gravitational field is thus accounted for in terms of the curvature of spacetime, but the electromagnetic field remains completely unrelated to the spacetime geometry. Consequently, it was quite natural to suggest that the electromagnetic field might also be ascribed to some property of spacetime, instead of being merely something embedded in spacetime.
Such a generalized differential geometry would describe both long range forces, and a new theory based on this geometry would constitute a unified field theory of electromagnetism and gravitation. In , Weyl proposed such a theory. The resulting geometry, called a Weyl geometry , is an intermediate geometric structure that lies between the conformal and Riemannian structures. This means that the metric field, locally represented by 8 , is invariant under parallel transport.
The coefficients of this unique symmetric linear metric connection are given by. While in Riemannian geometry the parallel transport of length is path independent , that is, it is possible to compare the lengths of any two vectors, even if they are located at two finitely different points, a vector suffers a path-dependent change in direction under parallel transport; that is, it is not possible to define the angle between two vectors, located at different points, in a path-independent way. Consequently, the angle between two vectors at a given point is invariant under parallel transport if and only if both vectors are transported along the same path.
In particular, a vector which is carried around a closed circuit by a continual parallel displacement back to the starting point, will have the same length, but will not in general return to its initial direction. For a closed loop which circumscribes an infinitesimally small portion of space, the rotation of the vector per unit area constitutes the measure of the local curvature of space. Consequently, whether or not finite parallel displacement of direction is integrable , that is, path-independent, depends on whether or not the curvature tensor vanishes.
According to Weyl, Riemannian geometry, is not a pure or genuine infinitesimal differential metric geometry, since it permits the comparison of length at a finite distance. However, in the Riemannian geometry described above, there remains a last distant-geometric [ferngeometrisches] element—without any sound reason, as far as I can see; the only cause of this appears to be the development of Riemannian geometry from the theory of surfaces. The metric permits the comparison of length of two vectors not only at the same point, but also at any arbitrarily separated points.
A true near-geometry Nahegeometrie , however, may recognize only a principle of transferring a length at a point to an infinitesimal neighbouring point , and then it is no more reasonable to assume that the transfer of length from a point to a finitely distant point is integrable, then it was to assume that the transfer of direction is integrable. Weyl wanted a metric geometry which would not permit distance comparison of length between two vectors located at finitely different points. In a pure infinitesimal geometry , Weyl argued, if attention is restricted to a single point of the manifold, then some standard of length or gauge must be chosen arbitrarily before the lengths of vectors can be determined.
Therefore, all that is intrinsic to the notion of a pure infinitesimal metric differential geometry is the ability to determine the ratios of the lengths of any two vectors and the angle between any two vectors, at a point. In the case of a pseudo-Riemannian structure such a gauge transformation leaves the light cones unaltered. Two metrics which are related by a conformal gauge transformation are called conformally equivalent. A conformal structure does not determine the length of any one vector at a point.
Weyl exploited these features of the conformal structure, and suggested that given a conformal structure, a gauge could be chosen at each point in a smooth but otherwise arbitrary manner, such that the metric 8 at any point of the manifold is conventional or undetermined to the extent that the metric. Moreover, the ratio of the lengths of two vectors located at different points is not determined even in a path-dependent way. As was pointed out earlier, this principle is satisfied in Riemannian geometry where the metric determines a unique symmetric linear connection, namely, the metric connection according to 9.
Evidently this fundamental principle of infinitesimal geometry is not satisfied for a structure which is merely a conformal structure, since the conformal structure only determines an equivalence class of conformally equivalent symmetric connections. The only condition imposed on the concept of congruent displacement of length is the following:. Theorem 4. The first term of the Weyl connection is identical to the metric connection 9 of Riemannian geometry, whereas the second term represents what is new in a Weyl geometry.
The Weyl connection is invariant under the gauge transformation. Therefore, the fundamental principle of infinitesimal geometry also holds in a Weyl geometry; that is, the metric structure of a Weyl geometry determines a unique affine connection, namely, the Weyl connection. Because of the latter property the formal characterization of the congruent displacement of length would be non-integrable, that is, path-dependent, in a Weyl geometry. Figure 6: In a Weyl geometry parallel displacement of a vector along different paths not only changes its direction but also its length.
Suppose physical spacetime corresponds to a Weyl geometry. This means that a twin who travels to a distant star and then returns to earth would not only discover that the other twin on earth had aged much more, but also that all the clocks on earth tick at a different rate. In that case the second term of the Weyl connection vanishes and 19 reduces to the metric connection 9 of Riemannian geometry. For example, the rate at which any clock measures time is a function of its history. But experience indicates otherwise. Atomic clocks define units of time, and experience shows they are integrably transported.
Thus, if we assume that the atomic time and the gravitational standard time are identical, and that the gravitational standard time is determined by the Weyl geometry, then the electromagnetic field tensor is zero. This means that in a Weyl geometry not only clocks would depend on their histories but also the masses of particles. For example, if two protons have different histories then they would also have different masses in a Weyl geometry.
But this violates the quantum mechanical principle that particles of the same kind—in this case, protons—have to be exactly identical. However, in it was still possible for Weyl to defend his theory in the following way. Since no detailed and reliable dynamical models were available at that time, Weyl could argue that there is no reason to assume that, for example, clock rates are correctly modelled by the length of a timelike vector. Weyl a, 67 said:.
At first glance it might be surprising that according to the purely close-action geometry, length transfer is non-integrable in the presence of an electromagnetic field. Does this not clearly contradict the behaviour of rigid bodies and clocks? The connection between the metric field and the behaviour of rigid rods and clocks is already very unclear in the theory of Special Relativity if one does not restrict oneself to quasi-stationary motion. Although these instruments play an indispensable role in praxis as indicators of the metric field, for this purpose, simpler processes would be preferable, for example, the propagation of light waves , it is clearly incorrect to define the metric field through the data that are directly obtained from these instruments.
Weyl elaborated this idea by suggesting that the dynamical nature of such time keeping systems was such that they continually adapt to the spacetime structure in such a way that their rates remain constant.
Lie algebra in physics
He distinguished between quantities that remain constant as a consequence of such dynamical adjustment , and quantities that remain constant by persistence because they are isolated and undisturbed. He argued that all quantities that maintain a perfect constancy probably do so as a result of dynamical adjustment. Weyl a, expressed these ideas in the following way:. What is the cause of this discrepancy between the idea of congruent transfer and the behaviour of measuring-rods and clocks?
I shall make the difference clear by the following illustration: We can give to the axis of a rotating top any arbitrary direction in space. This arbitrary original direction then determines for all time the direction of the axis of the top when left to itself, by means of a tendency of persistence which operates from moment to moment; the axis experiences at every instant a parallel displacement. The exact opposite is the case for a magnetic needle in a magnetic field.
Its direction is determined at each instant independently of the condition of the system at other instants by the fact that, in virtue of its constitution, the system adjusts itself in an unequivocally determined manner to the field in which it is situated. A priori we have no ground for assuming as integrable a transfer which results purely from the tendency of persistence. This circumstance shows that the charge is not determined by persistence, but by adjustment, and that there can exist only one state of equilibrium of the negative electricity, to which the corpuscle adjusts itself afresh at every instant.
For the same reason we can conclude the same thing for the spectral lines of atoms. The one thing common to atoms emitting the same frequency is their constitution, and not the agreement of their frequencies on the occasion of an encounter in the distant past. Similarly, the length of a measuring-rod is obviously determined by adjustment, for I could not give this measuring-rod in this field-position any other length arbitrarily say double or treble length in place of the length which it now possesses, in the manner in which I can at will pre-determine its direction.
The theoretical possibility of a determination of length by adjustment is given as a consequence of the world-curvature , which arises from the metrical field according to a complicated mathematical law. As a result of its constitution, the measuring-rod assumes a length which possesses this or that value, in relation to the radius of curvature of the field. Weyl could thus argue that it is at least theoretically possible that there exists an underlying dynamics of matter, such that a Weyl geometry, according to which length transfer is non-integrable, nonetheless coheres with observable experience, according to which length transfer appears to be integrable.
This relinquishment seems to have very serious consequences. While there now no longer exists a direct contradiction with experiment, the theory appears nevertheless to have been robbed of its inherent convincing power, from a physical point of view. For instance, the connexion between electromagnetism and world metric is not now essentially physical, but purely formal. For there is no longer an immediate connection between the electromagnetic phenomena and the behaviour of measuring rods and clocks. There is only an interrelation between the former and the ideal process which is mathematically defined as congruent transference of vectors.
Besides, there exists only formal, and not physical, evidence for a connection between world metric and electricity. As will be argued in more detail … there is, on the contrary, something to be said for the view that a solution of this problem cannot at all be found in this way. It was Lorentz who pointed out to Weyl that not only the world lines of light rays but also the world lines of material bodies are required for an intrinsic method of comparing lengths.
In addition, the latter presuppose quantum theoretical principles for their justification and therefore lie outside the relativistic framework because the laws which govern their physical processes are not known. Weyl c, abandoned his unified field theory only with the advent of the quantum theory of the electron. Dirac also argued that the time intervals measured by atomic clocks need not be identified with the lengths of timelike vectors in the Weyl geometry.
Prior to the works of Gauss, Grassmann and Riemann, the study of geometry tended to emphasize the employment of empirical intuitions and images of the three dimensional physical space. Physical space was thought of as having definite metrical attributes. The task of the geometer was to take physical mensuration devices in that space and work with them. The first satisfactory justification of the Pythagorean form of the Riemannian metric, although limited in scope because it presupposed the full homogeneity of Euclidean space, was provided by the investigations of Hermann von Helmholtz.
His analysis was thereby restricted to the cases of constant positive, zero, or negative curvature. As Weyl b points out, instead of a three-dimensional continuum we must now consider a four-dimensional continuum, the metric of which is not positive definite but is given instead by an indefinite quadratic form. Consequently, Weyl provided a reformulation of the space problem that is compatible with the causal and metric structures postulated by the theory of general relativity. But Weyl went further. More precisely, Weyl generalized the so-called Riemann-Helmholtz-Lie problem of space in two ways: First, he allowed for indefinite metrics in order to encompass the general theory of relativity.
It follows from these conditions that a Riemannian space possesses a definite symmetric linear connection—a symmetric linear metric connection [ 58 ] —which is uniquely determined by the Pythagorean-Riemannian metric. Weyl calls this:. The Fundamental Postulate of Riemannian Geometry: Among the possible systems of parallel displacements of a vector to infinitely near points, that is among the possible sets of symmetric linear connection coefficients, there exists one and only one set, and hence one and only one system of parallel displacement, which is length preserving. In his lectures [ 59 ] on the mathematical analysis of the problem of space delivered in at Barcelona and Madrid, Weyl sketched a proof demonstrating that the following is also true:.
Uniqueness of the Pythagorean-Riemannian Metric: Among all the possible infinitesimal metrics that can be put on a differentiable manifold, the Pythagorean-Riemannian metric is the only type of metric that uniquely determines a symmetric linear connection.
Weyl begins his proof with two natural assumptions. First, the nature of the metric should be coordinate independent. Definition 4. Weyl now states what he calls. Weyl emphasizes that the Postulate of Freedom provides the general framework for a concise formulation of. The Hypothesis of Dynamical Geometry: Whatever the nature or type of the metric may be—provided it is the same everywhere—the variations in the mutual orientations of the concrete microsymmetry groups from point to point are causally determined by the material content that fills space.
To assert this dynamical possibility does not require that the nature of the metric be specified. Next, Weyl points out that what has been provided so far is merely an explication of the concepts metric , length connection and symmetric linear connection. Some claim which goes beyond conceptual analysis has to be made, according to Weyl, in order to prove that among the various types of possible metrical structures that can be put on a differentiable manifold representing physical space, the Pythagorean-Riemannian form is unique.
Weyl suggests the following hypothesis:. Weyl shows that this hypothesis does in fact single out metrics of the Pythagorean-Riemannian type by proving the following theorem:. The nature of the metric field, that is the nature of the metric everywhere, is the same and is, therefore, absolutely determined. It reflects according to Weyl, the a priori structure of space or spacetime. In contrast, what is posteriori , that is, accidental and capable of continuous change being causally dependent on the material content that fills space, are the mutual orientations of the metrics at different points.
In the context of his group-theoretical analysis, Weyl b, p. I remark from an epistemological point of view: it is not correct to say that space or the world [spacetime] is in itself, prior to any material content, merely a formless continuous manifold in the sense of analysis situs; the nature of the metric [its infinitesimal Pythagorean-Riemannian character] is characteristic of space in itself, only the mutual orientation of the metrics at the various points is contingent, a posteriori and dependent on the material content.
At another place, Weyl a, Engl. Geometry unites organically with the field theory; space is not opposed to things as it is in substance theory like an empty vessel into which they are placed and which endows them with far-geometrical relationships. No empty space exists here; the assumption that the field omits a portion of the space is absurd.
According to Weyl, the metric field does not cease to exist in a world devoid of matter but is in a state of rest: As a rest field it would possess the property of metric homogeneity ; the mutual orientations of the orthogonal groups characterizing the Pythagorean-Riemannian nature of the metric everywhere would not differ from point to point. This means that in a matter-empty universe the metric is fixed. Consequently, the set of congruence relations on spacetime is uniquely determined. Since the metric uniquely determines the symmetric linear connection, the homogeneous metric field rest field determines an integrable affine structure.
Therefore, a flat Minkowski spacetime consistent with the complete absence of matter is endowed with an integrable connection and thus determines all hypothetical free motions. According to Weyl, there exists in the absence of matter a homogeneous metric field, a structural field Strukturfeld , which has the character of a rest field, and which constitutes an all pervasive background that cannot be eliminated. The structure of this rest field determines the extension of the spacetime congruence relations and determines Lorentz invariance. The rest field possesses no net energy and makes no contribution to curvature.
The contrast with Helmholtz and Lie is this: both of them require homogeneity and isotropy for physical space. From a general Riemannian standpoint, the latter characteristics are valid only for a matter-empty universe. Such a universe is flat and Euclidean, whereas a universe that contains matter is inhomogeneous, anisotropic and of variable curvature. On this manifold, Euclidean geometry turns out to be a special case resulting from a certain form of the metric. Weyl takes this general structure, the manifold structure, which has certain continuity and order properties, as basic, but leaves the determination of the other geometrical structures, such as the projective, conformal, affine and metric structures, open.
The metrical axioms are no longer dictated, as they were for Kant, by pure intuition. Weyl a, says:. We differentiate now between the amorphous continuum and its metrical structure. The first has retained its a priori character, [ 61 ] … whereas the structural field [Strukturfeld] is completely subjected to the power-play of the world; being a real entity, Einstein prefers to call it the ether. When interpreted physically, these mathematical structures or geometrical fields correspond, as Weyl says, to physical structural fields Strukturfelder.
Analogous to the electromagnetic field, these structural fields act on matter and are in turn acted on by matter. Weyl a, remarks:. I now come to the crucial idea of the theory of General Relativity. Whatever exerts as powerful and real effects as does the metric structure of the world, cannot be a rigid, once and for all, fixed geometrical structure of the world, but must itself be something real which not only exerts effects on matter but which in turn suffers them through matter. Riemann already suggested for space the idea that the structural field, like the electromagnetic field, reciprocally interacts with matter.
We already explained with the example of inertia, that the structural field [Strukturfeld] must, as a close-action [Nahewirkung], be understood infinitesimally. The manifold represents an amorphous four-dimensional differentiable continuum in the sense of analysis situs and has no properties besides those that fall under the concept of a manifold. The amorphous four-dimensional differentiable manifold possesses a high degree of symmetry. Because of its homogeneity, all points are alike; there are no objective geometric properties that enable one to distinguish one point from another.
This full homogeneity or symmetry of space must be described by its group of automorphisms , the one-to-one mappings of the point field onto itself which leave all relations of objective significance between points undisturbed. It is essentially for this reason that the real numbers are used for coordinate descriptions. Whereas the continuum of real numbers consists of individuals, the continua of space, time, and spacetime are homogeneous. Coordinates are introduced on the Mf [manifold] in the most direct way through the mapping onto the number space, in such a way, that all coordinates, which arise through one-to-one continuous transformations, are equally possible.
With this the coordinate concept breaks loose from all special constructions to which it was bound earlier in geometry. In the language of relativity this means: The coordinates are not measured, their values are not read off from real measuring rods which react in a definite way to physical fields and the metrical structure, rather they are a priori placed in the world arbitrarily, in order to characterize those physical fields including the metric structure numerically.
The metric structure becomes through this, so to speak, freed from space; it becomes an existing field within the remaining structure-less space. Through this, space as form of appearance contrasts more clearly with its real content: The content is measured after the form is arbitrarily related to coordinates. By mapping a given spacetime homeomorphically onto the real number space, providing through the arbitrariness of the mapping, what Weyl calls, a qualitatively non-differentiated field of free possibilities—the continuum of all possible coincidences—we represent spacetime points by their coordinates corresponding to some coordinate system.
Instead of thinking of the spacetime points as real substantival entities, and any talk of fields as just a convenient way of describing geometrical relations between points, one thinks of the geometrical fields such as the projective, conformal causal, affine and metric fields, as real physical entities with dynamical properties, such as energy, momentum and angular momentum, and the field points as mere mathematical abstractions. Spacetime is not a medium in the sense of the old ether concept. No ether in that sense exists here. Just as the electromagnetic fields are not states of a medium but constitute independent realities which are not reducible to anything else, so, according to Weyl, the geometrical fields are independent irreducible physical fields.
A class of geometric structural fields of a given type is characterized by a particular Lie group. A geometric structural field belonging to a given class has a microsymmetry group see definition 4. In relativity theory, this microsymmetry group is isomorphic to the Lorentz group and leaves invariant a pseudo-Riemannian metric of Lorentzian signature. Weyl a, says of this world structure:. However this structure is to be exactly and completely described and whatever its inner ground might be, all laws of nature show that it constitutes the most decisive influence on the evolution of physical events: the behavior of rigid bodies and clocks is almost exclusively determined through the metric structure, as is the pattern of the motion of a force-free mass point and the propagation of a light source.
And only through these effects on the concrete natural processes can we recognize this structure. The views of Weyl are diametrically opposed to geometrical conventionalism and some forms of relationalism. According to Weyl, we discover through the behavior of physical phenomena an already determined metrical structure of spacetime. The metrical relations of physical objects are determined by a physical field, the metric field, which is represented by the second rank metric tensor field. Contrary to geometric conventionalism, spacetime geometry is not about rigid rods, ideal clocks, light rays or freely falling particles, except in the derivative sense of providing information about the physically real metric field which, according to Weyl, is as physically real as is the electromagnetic field, and which determines and explains the metrical behavior of congruence standards under transport.
The metrical field has physical and metrical significance, and the metrical significance does not consist in the mere articulation of relations obtaining between, say, rigid rods or ideal clocks. The special and general, as well as the non-relativistic spacetime theories postulate various structural constraints which events are held to satisfy. When interpreted physically, these mathematical structures or constraints correspond to physical structural fields Strukturfelder. Analogous to the electromagnetic field, these structural fields act on matter and are, within the context of the general theory of relativity, in turn acted on by matter.
The mathematical model of physical spacetime is the four-dimensional pseudo-Riemannian manifold. Weyl c distinguished between two primitive substructures of that model: the conformal and projective structures and showed that the conformal structure, modelling the causal field governing light propagation, and the projective structure, modelling the inertial or guiding field governing all free fall motions, uniquely determine the metric.
That is, Weyl c proved. Under a conformal transformation. Weyl remarks after the proof:. If it is possible for us, in the real world, to discern causal propagation, and in particular light propagation, and if moreover, we are able to recognize and observe as such the motion of free mass points which follow the guiding field, then we are able to read off the metric field from this alone, without reliance on clocks and rigid rods.
As a matter of fact it can be shown that the metrical structure of the world is already fully determined by its inertial and causal structure, that therefore mensuration need not depend on clocks and rigid bodies but that light signals and mass points moving under the influence of inertia alone will suffice. The use of clocks and rigid rods is, within the context of either theory, an undesirable makeshift for two reasons.
Second, the concepts of a rigid body and a periodic system such as pendulums or atomic clocks are not fundamental or theoretically self-sufficient, but involve assumptions that presuppose quantum theoretical principles for their justification and thus lie outside the present conceptual relativistic framework. From the physical point of view, Weyl emphasized the roles of light propagation and free fall motion in revealing the conformal-causal and the projective structures respectively.
However, from the mathematical point of view, Weyl did not use these two structures directly in order to derive from them and their compatibility relation, the metric field. This is part of Cartan's theorem of highest weight. In summary, it provides a classification of the irreducible representations in terms of the weights of the Lie algebra.
Step two is composed of showing the following: Each dominant integral weight of a complex semisimple Lie algebra gives rise to a highest weight cyclic representation. Step one has the side benefit that the structure of the irreducible representations is better understood. Representations decompose as direct sums of weight spaces , with the weight space corresponding to the highest weight one-dimensional.
Repeated application of the representatives of certain elements of the Lie algebra called lowering operators yields a set of generators for the representation as a vector space. The application of one such operator on a vector with definite weight results either in zero or a vector with strictly lower weight. Raising operators work similarly, but results in a vector with strictly higher weight or zero.
The representatives of the Cartan subalgebra acts diagonally in a basis of weight vectors. All items in the above list work in the case of sl 2, C. What will actually be used below are irreducible representations of SL 2, C constructed from scratch, yielding via C1. For some semisimple Lie algebras, especially non-compact ones, it is easier to proceed indirectly via Weyl's unitarian trick instead of applying Cartan's theorem directly. See equation A1. Cartan's theorem is applied to sl 2, C , together with knowledge of its highest weights, yields a classification of the representations of so 3; 1 via A1.
An explicit construction of the irreducible representations of SL 2, C is presented, yielding explicit irreducible representations of sl 2, C , thus, via A1 , completing the task with the m , n representations of so 3; 1 as the final result. Representative matrices may be obtained by choice of basis in the representation space. An explicit formula for matrix elements is presented and some common representations are listed.
This is effected by taking the matrix exponential of the matrices of the Lie algebra representation. This results in the projective representations or two-value representations that are actually spin representations of the covering group SL 2, C. The Lie correspondence gives results only for the connected component of the groups, and thus the components of the full Lorentz that contain the operations of time reversal and space inversion are treated separately, mostly from physical considerations, by defining representatives for the space inversion and time reversal matrices.
According to the strategy , the irreducible complex linear representations of the complexification , so 3; 1 C of the Lie algebra so 3; 1 of the Lorentz group are to be found. A convenient basis for so 3; 1 is given by the three generators J i of rotations and the three generators K i of boosts. They are explicitly given in conventions and Lie algebra bases. The Lie algebra is complexified , and the basis is changed to the components of . One has the isomorphisms  [nb 17] Template:Equation box 1. The utility of these isomorphisms comes from the fact that all irreducible representations of su 2 , and hence see strategy all irreducible complex linear representations of sl 2, C , are known.
According to the final conclusion in strategy , the irreducible complex linear representation of sl 2, C is isomorphic to one of the highest weight representations. These are explicitly given in complex linear representations of sl 2, C. In A1 , all isomorphisms are C -linear the last is just a defining equality.
The most important part of the manipulations below is that the R -linear irreducible representations of a real or complex Lie algebra are in one-to-one correspondence with C -linear irreducible representation of its complexification see Strategy. The manipulations to obtain representations of a non-compact algebra here so 3; 1 , and subsequently the non-compact group itself, from qualitative knowledge about unitary representations of a compact group here SU 2 is a variant of Weyl's so-called unitarian trick.
The trick specialized to SL 2, C can be summarized concisely. Let V be a finite-dimensional complex vector space. The objects in the following list are in one-to-one correspondence. The correspondence is given via complexification of Lie algebras, via restriction to real forms, via the exponential mapping to be introduced , and via a standard mechanism also to be introduced for obtaining Lie algebra representations given group representations:. In this list, direct products groups or direct sums Lie algebras may be introduced if done consistently across per below.
The essence of the trick is that the starting point in the above list is immaterial. Both qualitative knowledge like existence theorems for one item on the list or properties like irreducibility and concrete realizations for one item on the list will translate and propagate, respectively, to the others. Here, the latter interpretation is intended. The representations for all Lie algebras and groups involved in the unitarian trick can now be obtained. The real linear representations for sl 2, C and so 3; 1 follow here assuming the complex linear representations of sl 2, C are known.
Explicit realizations and group representations are given later. The linearity properties follow from the canonical injection, the far right in A1 , of sl 2, C into its complexification. Template:Equation box 1. The labeling is sometimes in the literature 0, 1, 2, … or 0, 2, 4, … , but half-integers are chosen here to conform with the labeling for the so 3, 1 Lie algebra. Here the tensor product is interpreted in the former sense of A0. These representations are concretely realized below. Via the displayed isomorphisms in A1.
The notation D m , n is usually reserved for the group representations. These are, up to a similarity transformation , uniquely given by [nb 20]. These are explicitly given as . The approach in this section is based on theorems that, in turn, are based on the fundamental Lie correspondence. It is one-to-one in a neighborhood of the identity. The Lie correspondence and some results based on it needed here and below are stated for reference. If G denotes a linear Lie group i. The Lie correspondence reads in modern language, here specialized to linear Lie groups, as follows:.
Using the above theorem it is always possible to pass from a representation of a Lie group G to a representation of its Lie algebra g. It is explicitly computed using [nb 27] String Module Error: function rep expects a number as second parameter, received ". This, of course, holds for the Lorentz group in particular, but not all Lie algebra representations arise this way because their corresponding group representations may not exist as proper representations, i. If X is an element of so 3; 1 in the standard representation, then String Module Error: function rep expects a number as second parameter, received ".
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The subscript U indicates a small open set containing the identity. Its precise meaning is defined below. There are at least two potential problems with this definition. The first is that it is not obvious that this yields a group homomorphism , or even a well defined map at all local existence. The soundness of the tentative definition G2. This makes the map well-defined. The qualitative form of the Baker—Campbell—Hausdorff formula then guarantees that it is a group homomorphism, still for X small enough B. Technically, formula G2.
In detail,  String Module Error: function rep expects a number as second parameter, received ". It turns out that the result is always independent of the partitioning of the path. To see this, first fix a partitioning used in G3. Then insert a new point h somewhere on the path, say. Then, for any two given partitions of a given path, they have common refinement , their union. This refinement can be reached from any of the two partitionings by, one-by-one, adding points from the other partition.
For simply connected groups, the construction will be independent of the path as well, yielding a well defined representation. One deforms the path, a little bit at a time, using the previous result, the independence of partitioning. Each consecutive deformation is so small that two consecutive deformed paths can be partitioned using the same partition points. But any two pairs of consecutive deformations need not have the same choice partition points, so the actual path laid out in the group as progression is made through the deformation does indeed change.
This is a path from the identity to gh. Select adequate partitionings for p g , p h. This corresponds to a choice of "times" t 0 , t 1 , From a practical point of view, it is important that formula G2. It is a quotient of GL n , C see the linked article. The above construction relies on simple connectedness. The result needs modifications for non-simply connected groups per below.
The trace and determinant conditions imply . Thus topologically, . Formula G2. See the section spinors. Consider sl 2, C as a real Lie algebra with basis. From the relations String Module Error: function rep expects a number as second parameter, received ".
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The dot on the far right denotes path multiplication. An element of h may generally be written in the form String Module Error: function rep expects a number as second parameter, received ". This means that p A belongs to the full Lorentz group SO 3; 1. They can be and usually are written down from scratch.
The holomorphic group representations meaning the corresponding Lie algebra representation is complex linear are related to the complex linear Lie algebra representations by exponentiation. They can be exponentiated too. These are usually indexed with only one integer but half-integers are used here. The mathematics convention is used in this section for convenience.
Lie algebra elements differ by a factor of i and there is no factor of i in the exponential mapping compared to the physics convention used elsewhere. Let the basis of sl 2, C be  String Module Error: function rep expects a number as second parameter, received ". The action of SL 2, C is given by   String Module Error: function rep expects a number as second parameter, received ". The associated sl 2, C -action is, using G6. By employing G6. In general, if g is an element of a connected Lie group G with Lie algebra g , then  String Module Error: function rep expects a number as second parameter, received ".
In the case of the matrix q , one may write String Module Error: function rep expects a number as second parameter, received ". One finds String Module Error: function rep expects a number as second parameter, received ". As a corollary, since the covering map p is a homomorphism,the mapping version of the Lie correspondence G6. Refer to the commutative diagram. These are always representations since they are compositions of group homomorphisms. This is always a non-projective representation. For a Lie algebra g it reads .
There are three relevant cases. The center of any semisimple Lie algebra is trivial  and so 3; 1 is semi-simple and simple, and hence has no non-trivial ideals. A related fact is that if the corresponding representation of SL 2, C is faithful, then the representation is projective. Conversely, if the representation is non-projective, then the corresponding SL 2, C representation is not faithful, but is Accordingly, the corresponding projective representation of the group is never unitary.
Moyal Formulation of Quantum Mechanics
In fact, a connected simple non-compact Lie group cannot have any nontrivial unitary finite-dimensional representations. The kernel , ker U , of U is a normal subgroup of G. Since G is simple, ker U is either all of G , in which case U is trivial, or ker U is trivial, in which case U is faithful. In the case of the Lorentz group, this can also be seen directly from the definitions.
The representations of A and B used in the construction are Hermitian. This means that J is Hermitian, but K is anti-Hermitian. It is the SO 3 -invariant subspaces of the irreducible representations that determine whether a representation has spin. It cannot be ruled out in general, however, that representations with multiple SO 3 subrepresentations with different spin can represent physical particles with well-defined spin.
It may be that there is a suitable relativistic wave equation that projects out unphysical components , leaving only a single spin. To see if the dual representation of an irreducible representation is isomorphic to the original representation one can consider the following theorems:. Here, the elements of the Weyl group are considered as orthogonal transformations, acting by matrix multiplication, on the real vector space of roots. This follows from that complex conjugation commutes with addition and multiplication. The corresponding representations some R n or C n always exist for any matrix Lie group, and are paramount for investigation of the representation theory in general, and for any given Lie group in particular.
The left out part consists of functions of spacetime, differential and integral operators and the like.