The conventional statement of statistical determinism is that “the expectation values of all (Heisenberg)observables are determined by the expectation values of the observables at one time.” This requires that a full algebra of self-adjoint operators be in one-to-one correspondence with measurement procedures performed at one time. For instance, it requires that if two noncommuting observables pand q are defined at t = 0, there should exist a measurement procedure at t = 0 corresponding to p+q. No such procedure is known. The contrast between the positive assertion of the existence of certain laboratory procedures and the inability to describe them constitutes perhaps the weakest point of quantum mechanics. However, the conventional statement of statistical causality is shown to be untenable in a relativistic theory. This paper proposes a weaker form of causality which (1) uses measurements made within a truncated light cone rather than at one time for predictive purposes, and (2)which involves only strictly localized states, i.e., states which are vacuumlike outside a finite volume. Failure of the conventional causality statement implies that the set of quasilocal observables is not necessarily linear, i.e., if A and B are in a set, A+B is not necessarily in it. This remark may open the way to a systematic inquiry into the problems of associating laboratory procedures to self-adjoint operators.
The conventional statement of statistical determinism is that “the expectation values of all (Heisenberg)observables are determined by the expectation values of the observables at one time.” This requires that a full algebra of self-adjoint operators be in one-to-one correspondence with measurement procedures performed at one time. For instance, it requires that if two noncommuting observables pand q are defined at t = 0, there should exist a measurement procedure at t = 0 corresponding to p+q. No such procedure is known. The contrast between the positive assertion of the existence of certain laboratory procedures and the inability to describe them constitutes perhaps the weakest point of quantum mechanics. However, the conventional statement of statistical causality is shown to be untenable in a relativistic theory. This paper proposes a weaker form of causality which (1) uses measurements made within a truncated light cone rather than at one time for predictive purposes, and (2)which involves only strictly localized states, i.e., states which are vacuumlike outside a finite volume. Failure of the conventional causality statement implies that the set of quasilocal observables is not necessarily linear, i.e., if A and B are in a set, A+B is not necessarily in it. This remark may open the way to a systematic inquiry into the problems of associating laboratory procedures to self-adjoint operators.
We present in this paper a set of postulates for a physical system and deduce from these the main general features of the quantum theory of stationary states. Our theory is strictly operational in the sense that only the observables of the physical system are involved in the postulates. The collection of all bounded self-adj oint operators on a Hilbert space, which has previously been used as a mathematical model for the observables in quantum mechanics, satisfy the postulates, as do a variety of considerably more general mathematical structures. We show that for any two observables there exists a pure state of the system (this term being defined essentially as done by von Neumann and Weyl) in which they have different (expectation) values. Each observable has a spectral resolution, and a state of the system induces, in a natural way, a probability distribution on the range of spectral values of each observable. A number of observables are simultaneously observable if and only if they commute (although the product of two observables is not defined in general, a reasonable notion of commutativity can be introduced). Inasmuch as Hilbert space plays no role in our theory, our proofs are necessarily of a different character from the proofs of these results for the case of the system of all bounded selfadjoint operators. Actually, Hilbert space appears to be somewhat inadequate as a state space even for the latter system, in that there exist pure states of the system which cannot be represented in the usual way by rays of the Hilbert space. The postulates are partly algebraic and partly metric. The algebraic postulates require essentially that an observable can be multiplied by real numbers and raised to integral powers, and that any two observables can be added. It is assumed that the usual algebraic laws are satisfied so that (1) the observables can be treated like the elements of a linear space, and (2) the usual rules for dealing with polynomials in one variable with real coefficients remain valid when the variable is replaced by an observable. It is not assumed that two observables have a product. The metric postulates require that for each observable there be defined a kind of maximum numerical value, which plays the part of a norm, and has various properties in accord with its physical significance. While this norm is quite essential to the development of the theory, an interesting consequence of the theory is that the norm can be (uniquely) defined in a purely algebraic fashion. This shows that the objective features of a physical system,-the spectral values and probability distributions of the observables, and the pure states,-are completely determined by the algebra of observables, i.e., by the rules for addition, scalar multiplication, and powers, of observables.
Causality imposes strong restrictions on the type of operators that may be observables in relativistic quantum theories. In fact, causal violations arise when computing conditional probabilities for certain partial causally connected measurements using the standard non covariant procedure. Here we introduce another way of computing conditional probabilities, based on an intrinsic covariant relational order of the events, which differs from the standard one when these type of measurements are included. This alternative procedure is compatible with a wider and very natural class of operators without breaking causality. If some of these measurements could be implemented in practice, as predicted by our formalism, the non covariant, conventional approach should be abandoned.
Some of the problems connected with the interpretation of quantum mechanics are enumerated, in particular those related to some well known paradoxes and, above all, to the measurement process. We then show how the so called "Physics Laboratory Assumption" introduced in [1], which considers as "observables'' only the self-adjoint operators corresponding to existing measuring instruments, can propose a new perspective on the aforementioned problems and can replace the wavefunction collapse postulate.
[1] Luigi E. Picasso, "On the Concept of State in Quantum Mechanics: Another Way to Decoherence?'' Int. J. Theor. Phys. 62 (2), (2023)
The Hilbert-Space Description By the static description of a physical system we mean the rules that assign specified mathematical objects to the states and to the physical quantities of the system, and the prescriptions for calculating the probability distribution of the possible values of every physical quantity when the state of the system is given. In the usual Hilbert-space formulation of quantum mechanics, to each physical system is attached a separable Hilbert space (generally infinite-dimensional) over the complex field. To every physical quantity is associated a linear, self-adjoint, not necessarily bounded operator on. If one deals with a strictly quantum system, then the converse is also generally assumed: every self-adjoint operator on represents some physical quan- tity. The restriction “strictly quantum” is necessary: if the system retains some nonquantum (i.e. classical) feature, so that it requires the algorithm of so-called “superselection rules” (see Chapter 5), then there are self-adjoint operators on that do not represent physical quantities. It should also be stressed that even in the strictly quantum case, most of the self-adjoint operators actually do not represent “interesting” physical quantities: only a few of them represent physical quantities that are useful and meaningful for the description of the physical system (e.g., energy, momentum, position, angular momentum). Therefore, by asserting that every self-adjoint operator on corresponds to some physical quantity we mean there is no a priori impossibility of devising such a correspondence, but we do not claim that this correspondence is present in the real work of experimental physics.
In nonrelativistic quantum mechanics, the notion of the confined version of a system is proposed and discussed. Confined observables can turn out to be represented by just positive operator valued measures also when their nonconfined equivalents are represented by self-adjoint operators, and this nearly always happens for confined linear momentum observables. However, the self-adjoint operators of the nonconfined system can still be used to compute expected results and variances of confined observables. As a result, the precision with which some confined observables can be measured has limits which ultimately derive, through the Heisenberg commutation rules, from the covariance properties of nonconfined kinematical observables. Two conditions are proposed to characterize the approximation of a nonconfined system by a sequence of confined ones. A nonconfined system is deemed to be approximated by a sequence of confined systems if all probabilities determined by any observable of the nonconfined system with respect to nonconfined states can be approximated by confined counterparts and if the time evolution of any state of the nonconfined system can be approximated in any finite time interval by the time evolution of confined states. A rather general example of this situation is discussed.
Quantum physics can only make statistical predictions about possible measurement outcomes, and these predictions originate from an operator algebra that is fundamentally different from the conventional definition of probability as a subjective lack of information regarding the physical reality of the system. In the present paper, I explore how the operator formalism accommodates the vast number of possible states and measurements by characterizing its essential function as a description of causality relations between initial conditions and subsequent observations. It is shown that any complete description of causality must involve non-positive statistical elements that cannot be associated with any directly observable effects. The necessity of non-positive elements is demonstrated by the uniquely defined mathematical description of ideal correlations which explains the physics of maximally entangled states, quantum teleportation and quantum cloning. The operator formalism thus modifies the concept of causality by providing a universally valid description of deterministic relations between initial states and subsequent observations that cannot be expressed in terms of directly observable measurement outcomes. Instead, the identifiable elements of causality are necessarily non-positive and hence unobservable. The validity of the operator algebra therefore indicates that a consistent explanation of the various uncertainty limited phenomena associated with physical objects is only possible if we learn to accept the fact that the elements of causality cannot be reconciled with a continuation of observable reality in the physical object.