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Notes on a Space-Time Position Operator

Considering QM on one hand and SR on the other, there is a formal as well as conceptual clash at the level of kinematics: where SR constrains the kinematics by the Lorentzian invariant explicitly containing time, and in such a way to put it on the same level as a spatial coordinate, QM constrains kinematics with CCR not explicitly involving time. The usual formulation of QM in terms of canonically conjugate pairs seems glued to Hamiltonian formalism. Feynman's path integral formulation may yet be an out to this. One might argue that the concept of time enters indirectly through the momentum. But then, at the level of dynamics, time becomes simply the parameter for a one parameter group of unitary transformations. The introduction of a time operator in QM seems problematic. Among the problems is that it should destroy the equivalence between the Heisenberg and SchrX148dinger pictures. This equivalence would seem to depend on the centrality of time: time commutes with everything and can then be represented by a simple parameter. A lesson I learn from this is that the time of QM and the time of SR are conceptually and formally two different things.

In QM, the position operator Q has an eigenbasis |q> with eigenvalues q which correspond to Spatial Points. The formal concept of time does not enter QM as an obvious operator; if conceptually taken as an operator in the algebra of observables it is perforce "central", commuting, as it does, with all observables. Consider, conceptually, a necessary change that is imposed by the relativistic principle on a quantum theory: A relativistic observable cannot be simply "spatial", since a velocity dependent change of coR2|R0rdinate system, will change either the spectral values of such an operator or the expectation values of Q in some admissible state. Think of Lorentzian spatial contraction. For a time operator that may exist in any quantum theory, the argument applies. An essential part of the relativistic principle is that the Space-Time concept is the absolute observable concept, since it is this concept which is relativistically invariant. A space observable and a time observable should apparently be wedded into a single operator associated with an observable. This observable is then not one of "position" but one of "event". Moreover, unless it is obtained universally that for any space\-time separation that may be determined by any observer, and which separation corresponds to the determination of operators Q and T, that

          [Q, T]  =  0

it must be concluded that the "event point" which is to be specified by mathematically simultaneous eigenvalues of Q and T, cannot exist. As CCR coerces the classical nonlinear phase manifold to a linear complex Hilbert space, a lack of commutation between operators Q and T coerces the local Minkowski structure, (and its mapped image in a local region of a curved event manifold) to a linear complex indefinite Hilbert space. From the viewpoint of statistical mechanics, one simulates the uncertainty relations by hashing a classical phase space into a covering by neighborhoods of volume h/5S0nT4: similarly then, recover a conceptual semblance of a classical manifold by a cognate hashing of the spacetime manifold. One cannot take this image too seriously, since it is doubtful that the "absolute spacetime manifold" of SR or GR even exists. In QM, the future and the past of a system can only be determined statistically. With spacetime properly quantized, one can expect that the past and future local geometries and then also the connections of local geometries distributed on a presumed spacelike hypersurface can only be determined statistically. Spacelike related regions are classically disconected in a causal sense, as being outside of each other's causal domain. Even though the invariant speed c can be derived from quanta of space and time, yet can a possible quantum process exceed the speed of light, already even in QFT. A classical precursor to this objection to the global concept of "The Absolute Spacetime" may already be that the local approximation to the curved manifold of GR, given as Minkowski space does not exist within the manifold, but only as a tangent plane attached at a point. If this objection holds within and between say the Planck regime and the Heisenberg regime, it remains to explain the details of the apparent coherence of the universe. The explanation could be anticipated in the model of the limit in QM of "high quantum numbers" or "as h/ goes to zero". The coherence of the universe may also be provided from the GR point of view as a residue of the transluminesent possibilities of process necessary to a correct quantum theoretical accounting.

Without the convenience of the above commutator vanishing, as a generally valid principle or theorem, we are forced to accept the vector of a presumed Hilbert space to represent irreducibly, a "process". The transition from a state picture to a process picture seems inevitable when in possession of a time operator. Once this is transition is made, the quantum sense of world event is replaced with the sense of "eigenprocess". This subtly generalizes the rigidity of the Lorentz group acting on a spacetime composed of event points, while maintaining a causal structure or structure of distinction on the space of process which is the formal Hilbert space. Any possible events or processes become dynamical objects, inextricably bound with a concept of time, that are endowed with a causal relation.

Another lesson from SR, indeed also from GR, is that the time that we measure is told by clocks. An obvious point with some uncomfortable consequences, seen from the Newtonian viewpoint. Clocks are local cyclic contraptions; they are bounded in space, and finite temporally in their cycles and in their precision. This clock measured time is a local concept that can not in principle be wantonly extended thoughout space, in such a way as to provide a universally valid Newtonian time marching forward uniformly, at some given rate. The notion of simultaneity is not relativistically invariant, nor is the notion of the temporal ordering of distantly observed events. QM time is, however, Newtonian in all aspects. A clearly desirable break with with QM is the abandonment of its global nature, and therefore a retreat to a concept of local quantum theory of process, which is the subject of this investigation.

With regard to the everpresent quagmire of ontology in QM, it can be seen that in the context of the present theoretical model, the eigenprocesses of the Hilbert space would appear to be the correct models of physical process, that is to say that they possess a meaningful ontological status. It is not the illusory statistical spacetime that is the proper background for real" events, but rather the Hilbert space of process, and it is the Hilbert space that is endowed with the lightcone distinction specified by G(n) that distinguishes the processes as "spacelike", "lightlike", and "timelike". Moreover, since the a measurement must be considered as an interaction or confluence of process, "preceded" in a lab frame by at least two entering processes. The observed exiting processes although necessarily giving a definite result, do not generally remain in an "eigenstate" as QM would have it, since the eigenstates are actually eigenprocesses. The alleged eigenstates of energy are not stationary, even though they approach stationarity ar n becomes unbounded. Physically then, the collapse of the wave function simply does not happen.

There are two fundamental time concepts that must be distinguished. The first is the "time or clock phase" operator associated with a local system thought of as its own clock. This concept is very similar to the "internal periodic process" introduced by De\ Broglie, and whose straightforward relativistic transformation law, if it is defined by

This is the mathematical cognate of the "physical extension" in space for which we use the position operator. The second has to do with the time that we as macroscopic entities measure according to some local lab clock.

Therefore take as primitive, an operator G whose eigenvectors and eigenvalues are associated with world points. For simplicity, think of 1 spatial dimension and 1 temporal dimension. We expect a Minkowski-like structure to exist on our 2 dimensional spacetime. It is known, that an operator on an infinite dimensional Hilbert space may have a two dimensional spectrum [Putnam\ 1967]. As a rule, such operators are rather unruly. But we want the ability to specify world points with spectral points in a finite dimensional case where the spectrum will always be discrete. One can consider two possibilities immediately: 1) let the spectrum be complex or 2) let World points of the spectrum be on an embedded world line, which is of course 1\-dimensional. Extrapolation to higher spatial dimensions in case 1) seems a bit dicey (coR2|R0rdinating the spacetime content of the various spacetime operators for the other dimensions), but the extrapolation for case 2) seems not so tortured.

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Adopt case 2). Then, we need an additional operator G that expresses the Minkowski, light cone, or causal structure. The operator G when transformed by a Lorentz transformation that is a mere rotation in space will not alter the spectral set. A boost transformation will bring the notions of time dilation and Lorentz contraction into play in such a way that the expectation value of the invariant distance will not change. There will be processes that are classified as light cone processes. The additional operator is one that provides a causal structure on the processes space by having as its expectation value, the absolute Minkowski distance associated with a process from the origin of of the absolute 2\-dimensional spacetime. If %10|%00s> a process, is not zero, then \ =\ 0 implies that s is "expected to be on the light cone".

We approach world points given in terms of an invariant Minkowski length which determines a hyperbola, possibly degenerate, and a coR2|R0rdinate determining position on the hyperbola. If %10|%00s> is an eigenprocess, of G, we have a dispersion free process, on the light cone. For consistent connection of FCCR with QM in the limit n%10>%00%10I%00 where Q(%10I%00) is to be a purely spacelike position operator, spacelike is associated with a positive sign of and timelike with a negative sign:

       < 0 => %10|%00s> is a "timelike process,"
       > 0 => %10|%00s> is a "spacelike process,"

Geometrically, if %10|%00s> is timelike, the value of is the invariant hyperbolic distance from a hyperbola within the forward light cone, and the expectation values of G on all processes with the same are distributed on the hyperbola. Going to higher spatial dimensions, there are more operators G and still the single operator G to define the causal structure. The hyperbola generalizes as a higher dimensional hyperboloid of revolution about the time axis.

If %10|%00s> is spacelike, the same picture of a hyperbola obtains in the 2 dimensional spacetime, the hyperbola being rotated 90 degrees. In the case of higher spatial dimensions, it generalizes, however, to a pseudo\-Riemannian hypersphere.

The notions of "timelike", "spacelike" and "lightlike" are relative to the origin of an absolute spacetime coR2|R0rdinate system, and describe the expectation value of as a world point on the generalized conic section determined by . The general process %10|%00s> is associated with a probability distribution on the spectrum of G, which smears position over a possibly finite section of a hyperbola.

-1Indefinite Form G(n) Induces Minkowski Structure On A Space of Expectation Values-0 .fi .sp .PP The FCCR statement and the operators (2.8) are pure dimensionless. Introduce a length quantum %10l%005S10T4 and a time quantum %10t%005S10T4, so that they are related by the speed of light. .sp .nf c = %10l%005S10T4 / %10t%005S10T4 .fi .sp Redefine the operators Q(n) and P(n) by the scaling: .sp .nf Q(n) %10W1>W0 l%005S10T4 Q(n) P(n) %10W1>W0 %00h//%10l%005S10T4 P(n) where h/ is the reduced Plank constant, so that [Q(n), P(n)] = i h/ G(n) .fi .sp .PP A general vague rule for passing from a classical to a quantum statement is that classical variables get replaced with operators. Special relativistic principles center on the quadratic form in spacetime variables, .sp .nf q5S02T4 - c5S02T4T5S02T4 = +s5S02T4 .fi .sp An operator transcription of this we write by symbolically defining the operator T(n) so that: .sp .nf Q5S02T4(n) - c5S02T4T5S02T4(n) = %10l%005S10S02T4 G5S02T4(n) or T5S02T4(n) = (1/c5S02T4) ( Q5S02T4(n) - %10l%005S10S02T4 G5S02T4(n) ) .fi .sp Using lemma 8.3 and the fact that %10|%00q(n, k)> is an eigenvector of Q(n), we can calculate the matrix elements of T5S02T4(n). .sp .nf = %10t%005S10S02T4( q5S02T4(n, k) - (%10/%00(n-1))5S02T4 ) and for k =/ j, = %10t%005S10S02T4(n - 2)(-1)5S0(k-j)T4 .fi .sp A formal problem that arises immediately is that for a lightcone event, nonquantally, s5S02T4\ =\ 0 immediately implies that s\ =\ 0; the quantal variant of this, stated in terms of expectation values does not hold: vanishing of the expectation value of the square of some operator T5S02T4(n) associated with an observable, does not imply the vanishing of the expectation value of t(n). That \ =\ 0, has been proven, and \ = /\ 0. It is G(n) that immediately defines a cone structure in %10H%00(n), not G5S02T4(n). On the cone, the expectation value of G5S02T4(n) defines an uncertainty (%10D%00G(n))\ =\ %10/%00(n\-1), which is not "normalized". In SR, the invariant scalar %10/%00(\-s5S02T4), defines a measure of length or event separation in Minkowski space, while in a theory based on a simple operator replacement to produce the cognate operator equation, defines a measure of uncertainty in the "phase space", and also a measure of length or event separation in the Hilbert space. .sp .PP The notion of metric on a space of events has been lifted from the underlying position space of QM to the Hilbert space of events. This brings the Hilbert space formalistically closer to a spacetime construct and allows a greater topological flexibility in the nature of a quantal spacetime that is derived from prior algebraic structures; even the topological dimension of the derived spacetime is not a priori determinate. Each vector in the Hilbert space is associated with a singlet quantal world point. The set theory of world points should then be algebraically represented by the Clifford algebra of %10H%00(n) with G(n) as its inner product. It appears that these formal and conceptual differences will appear any time that a theory is constructed by a simple operator substitution as in the above. .sp .PP A second formal problem arises in a dynamical when a simple operator substitution is attempted. The algebraic equation expressing the relativistic invariant constructed from a position operator and a time operator, determines the time operator from the other two. Yet, in the context of Hamiltonian dynamics, the time operator should be chosen conjugate to a Hamiltonian operator. .nf WCMT Whoa! What really is the freedom in choosing a Hamiltonian? WCMT Choose a Hamiltonian. Presumably determine a conjugate time WCMT operator. Then satisfaction of the operator Lorentz condition WCMT selects vectors from the Hilbert space and is not satisfied for all WCMT vectors in the Hilbert space. WCMT What is the meaning of the selection? .fi The freedom to choose has been too severely delimited. We must loosen the statement of relativistic invariance, and associate the invariant Minkowski length with the expectation value of G(n), not G5S02T4(n) nor with some quadratic Casimir invariant. .PP It should also be remembered here that G(n) and Q(n) do not commute [Section IV]. .sp 4 -1Mapping to a Spacetime of Expectation Values-0 .sp .PP Consider a nonquantal two dimensional spacetime with signature chosen so that + is associated with space, - with time, and with a causal ordering induced by the lightcone structure. The group of transformations that preserves the causal ordering can be taken as that which preserves the bilinear form .sp .nf q5S02T4 - c5S02T4t5S02T4 = s5S02T4 .fi .sp where q has been taken as the spatial coR2|R0rdinate, t as the local time coR2|R0rdinate, c as the speed of light, and s as the invariant value of the bilinear form. It is not actually necessary in the case of a two dimensional spacetime that the group of causal automorphisms be the associated Lorentz group SO(1,\ 1) cross dilatations, but for higher dimensions this is true. The fact that we are considering two dimensions is supposed to be merely illustrative, and so we do not consider nonlinear causal automorphisms [Zeeman\ 1964]. .PP If s5S02T4\ =\ 0, the lightcone is described by a pair of straight lines intersecting at the origin. .sp .nf q = %10q%00ct .fi .sp if s5S02T4\ <\ 0, and fixed, q and t describe a pair of timelike hyperbolas within the forward and backward lightcones; .sp .ls 1 .nf t5S02T4 q5S02T4 %10------%00 |C-P4 %10---%00 = -1 (s/c)5S02T4 s5S02T4 .fi .ls 2 .sp if s5S02T4\ >\ 0, and fixed, q and t describe a pair of spacelike hyperbolas conjugate to the previous pair. .sp .ls 1 .nf t5S02T4 q5S02T4 %10------%00 |C-P4 %10---%00 = +1 (s/c)5S02T4 s5S02T4 .ls 2 .sp All the hyperbolas have lightcone asymptotes. .sp .nf Let (+s5S02T4) = , and = q, .fi .sp map processes of %10H%00(n) onto a 2\-dimensional spacetime whose "points" are expectation values, so that for any process %10|%00%10y%00> .sp .nf <%10y%00%10|%00Q(n)%10|%00%10y%00>5S02T4 - c5S02T4 <%10y%00%10|%00t(n)%10|%00%10y%00>5S02T4 = %10l%005S10S02T4 <%10y%00%10|%00G(n)%10|%00%10y%00> .fi .sp where the existence of an operator t(n) seems tacitly assumed. Actually, <%10y%00%10|%00t(n)%10|%00%10y%00>5S02T4 can be construed as a numerical quantity that is defined in terms of the other two numerical values that are derived from operators. Then %10|%00n,\ k> for k\ =/\ n\-1 are spacelike processes, %10|%00n,\ n\-1> is timelike, and %10|%00q(n,\ k)> and %10|%00p(n,\ k)> are lightlike. .PP Comparing general features of QM, SR and FCCR: Standard QM is a theory of action at a distance, with no explicit notion of causal propagation in its kinematics, and with a concept of a global absolute space. A system is defined by a Hamiltonian operator, and the Hilbert space is a space of states. QM states in the n\ %10>\ I%00 FCCR limit are presumably those for which <%10y%00%10|%00G(n)%10|%00%10y%00>\ =\ +1, that is, in FCCR context, normalized spacelike. .PP SR is causal in the sense of light cones, and an associated Cauchy problem, and references an absolute spacetime. .PP FCCR appears to have a natural causal structure built into the Hilbert space, whose elements are associated with spacetime events. Fuzzy, lightlike events can be taken as primary since the eigenvectors of Q(n) constitute a basis for the Hilbert space %10H%00(n); spacelike and timelike events are superpositions of lightlike events, similar in spirit as mentioned elsewhere to Feynman's path sum for the Dirac equation. Impure processes represented by density matrices are then distributions over spacetime events. The Hilbert space is then considered not so much a Hilbert space of "states" but rather a Hilbert space of "process", where meaning is dominated by transition amplitudes rather than expectation values. Process has a measure of size in the dimension n. The vectors %10|%00q(n,\ k)>, for example, represent lightlike processes associated with their eigenvalues q(n, k) of spatial position. Read a limit n%10>%00%10I%00 as, "'state' is the limit of large 'process'". A combinatorics of process is seemingly required to make connection with Feynman type kernels. .sp .nf Note that generally %10|%00%10|%005S02T4 = (q(n, k) - q(n, j))5S02T4 %10|%00%10|%005S02T4 = (p(n, k) - p(n, j))5S02T4 and for t(n) of the Harmonic Oscillator %10|%00%10|%005S02T4 = (t(n, k) - t(n, j))5S02T4 All the eigenvectors of Q(n), P(n) and t(n) are G(n)\-null, and it is an inner derivation by G(n) that provides measures of a physical distance associated with pairs of G(n)\-null eigenvectors. .fi .sp 4 .ls 2 .bp .sp -1Processes With Minimal and Maximal Uncertainty In G(n)-0 .PP It would be of some interest to say something about uncertainty of G(n), since it corresponds to an uncertainty in specifying the hyperbola associated with an invariant Minkowski length. .sp .ls 1 .nf (%10D%00G(n))5S02T4 = - 5S02T4 where the expectation value is defined as, %10=%00 / which in %10|%00n, k> basis is - n%10|%00%10|%005S02T4 = %10---------------------%00 %10|%00%10|%005S02T4 = 1 - n %10-------------%00 and %10=%00 / which in %10|%00n, k> basis is + n(n-2) %10|%00%10|%005S02T4 = %10------------------------------%00 %10|%00%10|%005S02T4 = 1 + n(n-2) %10-------------%00 Therefore (%10D%00G(n))5S02T4 = - 5S02T4 %10|%00%10|%005S22T04 %10|%00%10|%005S22T04 = n5S22T04 %10-------------%00 [ 1 - %10-------------%00 ] If we define %10|%00%10|%00 r %10= ------------%00 5S01/2T4 which expresses the weighting of %10|%00n, n-1> in %10|%00s> (r is always real and less than 1.) then, = 1 - n r5S02T4 = 1 + n(n-1)r5S02T4 (%10D%00G(n)) = n r (1 - r5S02T4)5S01/2T4 If r = 0, then = 1 = 1 (%10D%00G(n)) = 0 the kind of values one would expect in QM. If r =/ 0 we can find minimal uncertainty processes with respect to r. Eliminating r to express (%10D%00G(n)) in terms of : (%10D%00G(n)) = [ (1 - ) ( - (-n+1)) ]5S01/2T4 the geometric mean of the distances from end points of the range of . Consider the function f(z) = [(a - z)(z - b)]5S01/2T4 for the range -b %10T%00 z %10T%00 a. f(z) = 0 only when z = a or z = b. If the first derivative is zero, the critical point z = (1/2)(a + b) is determined. At this point f(z) has the value (1/2)(a - b) The second derivative is f"(z) = 1/f(z)[ -1 - (f'(z)/2f(z))(b + a -2z)] .fi .ls 2 .sp The second term vanishes at the critical point, so the sign of the second derivative at the critical point depends only on f(z). The value is then also \-1/f(z). That is, at a critical point, .sp .nf f"(z) = - 2/n which is always negative. From the expression above for (%10D%00G(n)) the critical point is at = 1 - n/2 At this point then (%10D%00G(n)) = n/2 which is always positive. Therefore, for expectation values at the the critical point, the uncertainty (%10D%00G(n)) is at a local -1MAXIMUM-0. For processes with maximum uncertainty in G(n) 5S1criticalT4 = 1 - (%10D%00G(n))5S1maxT4 There are absolute minimal zero values for uncertainty in G(n) at the end points of the range of . For the subranges of classifying the processes: .sp .KS .nf (spacelike r=0) = +1 => (%10D%00G(n)) = 0 (timelike r=1) = -(n-1) => (%10D%00G(n)) = 0 Also (lightlike r=1/%10/%00n) = 0 => (%10D%00G(n)) = %10/%00(n-1) .KE .fi .sp This last result is a generalization for arbitrary %10|%00%10y%00> such that <%10y%00%10|%00G(n)%10|%00%10y%00> = 0, of the special case computable from lemma 8.3 for the %10|%00q(n,\ k)> and also %10|%00p(n,\ k)>. .sp .PP For large n, we know that the spectral radius of Q(n) and P(n) grows as %10/%00n, and that that of G(n) grows as n. The algebra of observables is generated by polynomials in Q(n) and P(n). The algebra is essentially generated by the polynomials of order %10T%00 n in that, any polynomial of order greater than n is equivalent as a matrix to a polynomial of order less than or equal to n. [Cf. section VI]. Also, any polynomial that commutes with all the polynomials of order %10T%00 n or less is a multiple of the identity. [This actually needs to be proved.] We also want to consider the limit of the algebra of observables as n%10>%00%10I%00, knowing that in QM at least one of Q and P must be unbounded and therefore have an unbounded spectral radius. We want to lose as few of the observables as possible in taking the limit. If the spectral radius of Q(n) grows as %10/%00n, then the spectral radius of a power, Q5S0kT4(n) grows as n5S0k/2T4. In particular, if we do not wish to damp the growth of the spectral radius of Q5S0nT4(n) but do not care about higher powers, then the limit that should be considered is: .sp .ls 1 .nf lim %10--------%00 n5S01/4T4 [See section III and the decomposition of %10H%00(n) under the action of the u(n) algebra of observables] 5S1criticalT4 -> 1 5S1criticalT4 %10-------------%00 -> -(1/2) n and (%10D%00G(n))5S1criticalT4 %10-----------------%00 -> 1/%10/%002 n .sp .ls 2 .fi .sp 2 .nf -1Central and Noncentral Time in Quantum Theories-0 .fi .sp .PP In both CM and QM time appears as the parameter of the one parameter dynamical group that describes the evolution of the physical system. One can construct a homogeneous Hamiltonian formalism for CM in which the spatial coR2|R0rdinates are on the same footing as a time coR2|R0rdinate, but in standard QM there is an essential algebraic difference between a spatial coR2|R0rdinate and a time coR2|R0rdinate: a spatial variable is represented in QM as an Hermitean operator in an algebra of observables. A spatial or position variable cannot commute with all other observables; in fact, CCR itself denys this possibility. If the time variable is to be represented by an operator in the algebra of observables, it is one that always commutes with all other observables and therefore is in the center of the algebra. If the operator algebra representation is irreducible, then the time operator is a multiple of the identity, and has only one eigenvalue. If the representation is reducible, then we can consider a dynamical direct product fibre bundle with the time manifold as base space, the usual algebra of observables as fibre, a "largest" unitary group of transformations acting on operators as the group of the bundle, and dynamical trajectories as cross sections of the bundle. In the direct product, a time operator is then orthogonal to the other operator observables. Consider a Hamiltonian operator in QM which does not explicitly depend on time. Using the SchrX148dinger equation, the initial apparent loss of generality in the separation of variables assumption is indeed apparent and a general solution to the SchrX148dinger equation can be written as a linear superposition of energy eigenstates with time dependent coefficients: .sp .nf %10|%00%10y%00(t)> = %10S%005S1kT4 %10a%005S1kT4 exp( iE5S1kT4t/h/ ) %10|%00E5S1kT4> %10|%00%10y%00(0)> = %10S%005S1kT4 %10a%005S1kT4 %10|%00E5S1kT4> where %10|%00E5S1kT4> is an eigenstate of the Hamiltonian with eigenvalue E5S1kT4. .fi .sp The Hermitean position and momentum operators Q and P also have, modulo distribution technicalities, complete orthonormal eigenbases with nondegenerate real eigenvalues, so that .sp .nf Q %10|%00q5S1kT4> = q5S1kT4 %10|%00q5S1kT4> P %10|%00p5S1kT4> = p5S1kT4 %10|%00p5S1kT4> An arbitrary state %10|%00%10y%00> can be expanded on either basis set. %10|%00%10y%00> = %10S%005S1kT4 %10b%005S1kT4 %10|%00q5S1kT4> or %10|%00%10y%00> = %10S%005S1kT4 %10a%005S1kT4 %10|%00p5S1kT4> Q and P are related by a Fourier transform whose matrix elements are represented by . %10|%00p5S1kT4> = %10S%005S1jT4 %10|%00q5S1jT4> so %10b%005S1jT4 = %10S%005S1kT4 %10a%005S1kT4 and %10|%00%10y%00> = %10S%005S1jT4 (%10S%005S1kT4 %10a%005S1kT4 ) %10|%00q5S1jT4> .fi .sp These relations do not depend on the existence of the Fourier transform but only on the existence of a transformation relating two basis sets. As usual, the preservation of <%10y|y%00> is the unitarity condition for the transformation. .PP We want to see what modifications of formalism and interpretation are needed if QM is to embrace a non\-central time. .PP Suppose there exists T(n) a noncentral time operator that is Hermitean in a *\-algebra of operators acting on %10H%00(n). Assume also that the eigenvalues of T(n) are nondegenerate. There is then an orthonormal basis %10|%00t5S1kT4> of %10H%00(n) such that .sp .nf T(n) %10|%00t5S1kT4> = t5S1kT4 %10|%00t5S1kT4> .fi .sp With the existence of an eigenbasis for a Hamiltonian we can write for any element of %10H%00(n) .sp .nf %10|%00%10y%00> = %10S%005S1jT4 (%10S%005S1kT4 %10a%005S1kT4 ) %10|%00E5S1jT4> or %10|%00%10y%00> = %10S%005S1jT4 (%10S%005S1kT4 %10b%005S1kT4 ) %10|%00t5S1jT4> .fi .sp Clearly the time solution of the SchrX148dinger equation is a special case of this form, where for any j, .sp .nf %10a%005S1kT4 exp( iE5S1kT4t5S1jT4/h/ ) = %10S%005S1iT4 %10a%005S1iT4 .fi .sp The condition that time and energy commute, leads to the possibility of writing the time dependent solution for an energy eigenstate as .sp .nf %10|%00E5S1kT4(t)> = exp( iE5S1kT4t/h/ ) %10|%00E5S1kT4> .fi .sp where any value of t is construed as an eigenvalue of the time operator. Otherwise, we would have to generalize such states to those which are expressed as a distribution over the eigenstates %10|%00t5S1kT4> above. The expression of a time dependent solution then actually becomes and expression of transformation between the basis %10|%00t5S1kT4> and the basis %10|%00E5S1kT4>. But if the operators H(n) and t(n) are fixed, this is an identity and if construed as a dynamical expression, expresses a "frozen" dynamics: we have no dynamical group. .PP A thawing can occur if the transformation between the two basis is allowed to be for instance dependent on a parameter, which might be taken to be relativistic proper time s, so that .sp .nf t(n, s) = %10U%00%10!%00(n, s) H(n) %10U%00(n, s) .fi .sp which defines a time operator in terms of the Hamiltonian and a one parameter group of transformations whose parameter is proper time. If %10U%00(n,\ s) is unitary, t(n,\ s) has the same spectrum as H(n) for all values of s. (The actual %10U%00(n,\ s) depends on the Hamiltonian.) A generalization of this relation so that Sp( t(n, s) ) is a linear function of Sp( H(n) ) is easy: With A, B %10e R%00, define .sp .nf t(n, s) = A %10U%00%10!%00(n, s) H(n) %10U%00(n, s) + B I(n) .fi .sp Using G(n) or even some arbitrary operator perhaps depending on s instead of I(n) then also springs to mind. Then the eigenvalues of t(n, s) are each independent functions of the eigenvalues of H(n) and also of s. However, using the simpler relation we have identically .sp .nf [H(n), t(n, s)] = - [%10U%00%10!%00(n, s)[H(n), %10U%00(n, s)], t(n)] .fi .sp What should this commutator be equal to? It should be related to FCCR. .sp The natural definition of a proper time associated with a state %10|%00%10y%00> is .sp .nf s5S02T4(%10y%00) %10=%00 <%10y%00%10|%00G(n)%10|%00%10y%00> .fi .sp It must be related to the expectation values of the operator Q and a local time operator in a way that expresses the relativistic invariance of s. .PP Lorentz transformations should act on the algebra so that Q(n) and ct(n,\ s) are components of a spinor operator of SU(1,\ 1), transforming according to the defining representation, so that for R an element of the group SU(1,\ 1) acting on %10H%00(n) [See Messiah\ 1966], II. p. 1075 in terms of the "Standard Components" of the spinor operator, we have: .sp .nf P5S1RT4 T5S1kS0(1/2)T4 P5S1RS0-1T4 = %10S%005S1jT4 T5S1jS0(1/2)T4 D5S1jkS0(1/2)T4(R) or in term of the generators J5S1%10q%00T4 and J5S13T4, of SU(1, 1) [This is actually for SU(2)] [J5S1%10q%00T4, T5S1qS0(w)T4] = %10/%00( w(w+1) - q(q%10q%001) ) T5S1q%10q%001S0(k)T4 [J5S13T4, T5S1qS0(w)T4] = q T5S1qS0(w)T4 .KS .ls 2 .nf Specifically for w=(1/2), and q=(+1/2) or q=(-1/2) and the two standard components T5S1+S0(1/2)T4 and T5S1-S0(1/2)T4: [J5S1%10q%00T4, T5S1+%10_%00S0(1/2)T4] = T5S1%10q%00S0(1/2)T4 [J5S1%10q%00T4, T5S1%10q%00S0(1/2)T4] = 0 [J5S13T4, T5S1%10q%00S0(1/2)T4] = %10q%00(1/2) T5S1%10q%00S0(1/2)T4 .KE .KS .ls 2 .nf or [J5S11T4, T5S1%10q%00S0(1/2)T4] = (1/2) T5S1+%10_%00S0(1/2)T4 [J5S12T4, T5S1%10q%00S0(1/2)T4] = (1/2) T5S1+%10_%00S0(1/2)T4 [J5S13T4, T5S1%10q%00S0(1/2)T4] = %10q%00(1/2) T5S1%10q%00S0(1/2)T4 .KE .fi .sp Actually all we really need is one hyperbolic rotation (This is presumably a two dimensional spacetime.) that is in the kinematical invariance group U5S1gT4(n). Similarly for P(n) and H(n). (contragrediently?) So the expectation values as a timelike vector (,\ c) transform according to the matrix: .sp .ls 1 .nf %10|%00cosh %10o%00 sinh %10o%00%10|%00 r5S1hT4 = %10|%00 %10|%00 %10|%00sinh %10o%00 cosh %10o%00%10|%00 with tanh(%10o%00) = v/c and v is an expected 3\-velocity. Relativistically, (s/)5S02T4 = 1 - v5S02T4/c5S02T4 or v/c = %10/%00(1 - s5S02T4/5S02T4) .fi .ls 2 .sp .sp .PP In the context of either SR or GR, the metric tensor g5S1%10m%00%10n%00T4(x) defines both a causal structure and a measure on spacetime. Locally, in GR, gauge transformations are performed to expose SR in tangent spaces. U0

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Created: August 1997
Last Updated: May 28, 2000
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