Abstract
We present a brief exposition of the orthodox axiomatic approach to q.m., indicating the relation to the text-book approach. We explain why the usual axioms force a change of logic. We then explain the attempts to derive the Hilbert space and the probability interpretation from a new type of ‘and’ and ‘or’ or a new type of ‘if’ and ‘not’. Included are the Birkhoff-von Neumann, Jauch-Piron, and quantum logic approaches, together with an account of their physical and mathematical obscurities.
Instead of entering the labyrinth of subsequent developments, which seek new algebraic structures while accepting the old physical motivation, we present an exposition of the structured-time interpretation of q.m., which seeks a new physical motivation.
We saw in Chapter VB that, with a tilt in the arrow of time, the solutions of the many-body equations of motion are intrinsically non-unique. In Chapter VIA we had indicated how this non-uniqueness relates to a change in the logic of time. We now explain how the resulting changes in the logic and structure of time lead to a new type of ‘if’ and ‘not’, of the kind required by q.m., while escaping from the criticism which applies to the earlier ‘quantum logic’ approaches.
We briefly indicate the analogy between this logic and the temporal logic required for the formal semantics of parallel-processing languages like OCCAM, and distinguish the structured-time interpretation from the superficially similar many-worlds interpretation and the transactional interpretation of q.m.
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Notes and References
Though we do not enter into this question here, a structured time is also closer to the time of daily experience.
For those who have only vaguely heard of measure theory, the significance of Borel sets is that it always makes sense to speak of the measure of such a set with respect to a regular Borel measure! The significance of a (finite) regular Borel measure is that it corresponds (both ways) to an integral which will (i) integrate continuous functions, and (ii) provide a finite value for the integral of a continuous function which ‘vanishes at infinity’, i.e., is less than the proverbial epsilon outside a compact set. The formal definition of a Borel set in a topological space Xis that it is an element of Bx, the smallest a-algebra which contains all open sets.
P. R. Halmos, Introduction to Hilbert space and the Theory of Spectral Multiplicity, Chelsea, New York, 1951. The whole book is devoted to the proof of this theorem.
From Dirac’s classic treatise on Quantum Mechanics,as cited by J.S. Bell, Speakable and Unspeakable in Quantum Mechanics (Cambridge: University Press, 1987), p 40.
For the difficulties in extending von Neumann’s collapse postulate to observables with continuous spectra, and for a new formulation see M.D. Srinivas, ‘Collapse postulate for observables with continuous spectra’, Commun. Math. Phys., 71, 131–58 (1980).
E. P. Wigner, Phys. Rev., 40, 749 (1932).
E. P. Wigner, in: Perspectives in Quantum Theory, W. Yourgrau and A. van der Merwe (eds), MIT Press, Cambridge, Mass., 1971.
The other possibility, pursued for instance in stochastic quantization, is to change the axiomatic basis of q.m., so that joint probability distributions do exist even for canonically conjugate variables.
For the measure-theoretic foundations of probability theory, as enunciated by Kolmogorov, see, for instance, P. Halmos, Measure Theory, D. Van Nostrand, New York, 1950. To change from sets to sentences, see, e.g., J. R. Lucas, The Foundations of Probability, Clarendon, Oxford, 1957.
The logico-algebraic approach was initiated by G. Birkhoff and J. von Neumann, Ann. Math., 37, 823 (1936). For further developments of this theory, see V.S. Varadarajan, Geometry of Quantum Theory, Vol. 1, Van Nostrand, Princeton, N.J., 1968, and C. A. Hooker (ed) The Logico AlgebraicApproach to Quantum Mechanics, D. Reidel, Dordrecht, 1975.
A ‘subspace’ will always mean a closed linear manifold.
One has to be a little careful in applying the analogy to implication. The sentential connective should not be confused with the relation of implication. Thus, it makes sense to speak of (B’’A),but not of a_ (b _a).
See G. Kalmbach, Hilbert Lattices,World Scientific, Singapore, 1987, for a counter-example, which however, does not apply to probability measures.
Those interested in probing further may like to be discouraged by the text of S. Lang, Algebra, Addison-Wesley, Reading, Mass., 1965.
C. K. Raju, ‘Quantum mechanics and the microphysical structure of time’ [1990]. In: R. Nair (ed) In Search of Quantum Reality (to appear).
This is true directly for charged particles, and indirectly, at least, for all particles which interact in any way with charged particles.
The whole debate over ‘hidden variables’ is related to the metaphysical assumption (of linear time) that ‘reality is unambiguous’. This assumption breaks down with branching time.
The terminology derives from temporal logic.
See, e.g., K.R. Apt (ed), Logics and Models of Concurrent Systems,Springer, Berlin, 1984, NATO ASI Series in Computers and Systems Science, Vol. 13. The moral of the story is that one should link non-locality to CSP rather than ESP!
The usual definition of a random variable as a measurable function assumes the possibility of such an in-principle specification.
N. Rescher and A. Urquhart, Temporal Logic ( Wien: Springer, 1971 ).
This does not lead to any paradoxes, as discussed in detail in Chapter VB. In terms of the analogy to the formally similar but more pragmatic CSP, the ALT construct of OCCAM is non-deterministic, and the contingent ‘present’ is what accounts for the current practical difficulty in constructing a debugger for parallel processors. Incidentally, in India, this difficulty has been currently resolved by abandoning CSP and adopting a queuing model in the PARAS system software developed by C-DAC for use with parallel FORTRAN, and C, which have some constructs resembling the PAR and ALT of OCCAM. The empirical significance of a contingent past is discussed later on.
Traditionally, a proposition is what a sentence expresses or means. Formally, ‘meaning’ may be deemed to have been assigned to a sentencep by specifying the set of ‘worlds’, Ip I, in which the sentence is true, so that one knows when the sentence is true and when it is false. Since we shall be using different notions of ‘world’, we should have, strictly, distinguished between statements, truth-functional propositions and quasi truth-functional propositions. We shall, however, not retain this difference explicitly, in the hope that the type of entity being considered would be clear from the context.
Bas C. Van Fraassen, in: Hooker (ed) Contemporary Research in Foundations and Philosophy of Quantum Theory, D. Reidel, Dordrecht, 1973, p 84.
Strictly speaking, a state should be identified with a convex combination of the probability measures induced by each q.t.f. world in the collection. Thus, to each q.t.f. world w in a state, one must also assign a number t(w),0t(w)1, such that E t(w) = 1, the sum being taken over all q.t.f. worlds in the state.
The selection function is, in fact, a projection: fp o fp = fp, and could, under the circumstance (25), quite literally be identified with a projection operator in Hilbert space.
W. L. Harper, R. Stalnaker, and G. Pearce (eds), Ifs, D. Reidel, Dordrecht, 1981.
See Harper et al, Ref. 27.
I will use statements and propositions interchangeably; see note 23.
Ref. 15, Theorems 1, and 2.
Ref. 15, Theorem 5, equivalence of (i) and (v).
Ref. 15.
J.C.T. Pool, (reproduced in Hooker, Ref. 24 above). In the next paper in Hooker (ed), Pool also considers the possible significance of semi-modularity.
J. G. Cramers, Rev. Mod. Phys., 58, 647 (1986).
See van Fraassen, Ref. 24.
As a specific example, consider van Fraassen’s assumption (2), in Ref. 24, corresponding to Pool’s Axiom 1.3, in Ref. 33. This says that Op Oq implies Op 0 q. That is, if the necessity ofp implies the necessity of q,then the possibility of p implies the possibility of q. I do not know of any traditional modal logic in which this is true.
C. K. Raju, Int. J. Theor. Phys., 20, 681 (1980).
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Raju, C.K. (1994). Quantum-Mechanical Time. In: Time: Towards a Consistent Theory. Fundamental Theories of Physics, vol 65. Springer, Dordrecht. https://doi.org/10.1007/978-94-015-8376-3_10
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