The question that different mechanical quantities of quantum mechanics have definition values simultaneously is studied. As a result, the question depends on the states of quantum system. The question relates whether the operator of mechanical quantity can commute,but they don't form the causal relation.
Despite its power as the conceptual basis for a huge range of physical phenomena in atomic and subatomic physics, quantum mechanics still suffers from a lack of clarity regarding the physical meaning of its fundamental theoretical concepts, such as those of quantum state and of quantum theoretical quantities or variables , dealt with by the known mathematical-theoretical rules. These concepts have generally been considered as not giving a direct description of physical systems, for they do not correspond to what is usually understood by `physical states' or `physical properties', notably characterized by definite numerical values such as those obtained from measurement. The situation has been tentatively expressed in terms of various `interpretations', conceived not only with regard to the physical meanings of mathematical quantities, but also in connection with philosophical statements about `physical reality'. The question of whether or not quantum theoretical quantities describe definite physical systems existing in nature has therefore generally been considered as beyond the possibilities of physics, because of the definitions that are commonly taken for `physical state' and `physical quantity'. We sketch the main ideas on this problem and propose a possible way out of the puzzle, in terms of an extension of the meaning given to the concepts of the physical state and physical quantity of a system, which would allow one, without any theoretical change in quantum mechanics, to speak consistently of real quantum systems as having definite physical properties .
In classical physics, the ‘state’ of a particle, at any particular time, is taken to involve a definite position in space and a definite motion; and the state of a system of particles is taken to involve a definite position in space and a definite motion for each of the particles. The situation in relation to quantum mechanics is completely different. As in classical physics, calculations can be made concerning the position and motion of a particle. However, this representation is not by means of symbols for scalar or vector quantities: it is by a more complex mathematical object. This quantum state does not indicate or represent position or motion in any direct way: what it does in substance is to enable calculation of the probability of the particle in question being found to have a position or momentum if an appropriate measurement is made.
In classical mechanics, as we know, all observables commute with each other and therefore belong to the same “commuting set.” This means that – in principle – we can measure all observables simultaneously. Then, the state manifests itself as a collection of properties (of values of observables) (see Sec. 1.1), and as such it is observable. On the other hand, quantum-mechanical observables may not commute (see Subsec. 2.1.5). As a consequence, the state is not just a collection of observable values (see also Sec. 2.3) so that we may ask ourselves whether it is observable or not, and therefore raise the following question: What is the nature of the wave function? Is it simply a mathematical tool, i.e. does it only represent the sum of all that we can know about a system? Or does it have an ontological status? Since the early days of quantum mechanics, this question has been very popular. As a first attempt, de Broglie tried to interpret the wave function as describing, in classical terms, a wave field that envelopes a classical particle (see also Subsec. 16.3.2). This theory should predict the existence of empty waves, but several experiments have shown negative results (see also the end of Sec. 9.6). More recently, many physicists have tried to propose some methods in order to measure the wave function. Here we shortly report on some of these proposals and discuss to what extent the quantum state can be measured.
Quantum mechanics predicts the joint probability distribution of the outcomes of simultaneous measurements of commuting observables, but, in the state of the art, has lacked the operational definition of simultaneous measurements. The question is answered as to when the consecutive applications of measuring apparatuses give a simultaneous measurement of their observables. For this purpose, all the possible state reductions caused by measurements of an observable is also characterized by their operations.
Wave--particle duality is a cornerstone of quantum mechanics, traditionally formulated under definite causal order. We investigate how complementarity is modified when the temporal order of operations is coherently superposed, as in the quantum switch. We show that no universal, state-independent, linear additive complementarity relation exists that combines spatial coherence, path distinguishability, and causal coherence within a common unit-normalized bound. This result follows from explicit extremal constructions in the quantum-switch family: although the three quantities are derived from the same global switch state, the standard spatial duality relation can be saturated while the reduced control coherence is simultaneously maximal. Tracing out the order qubit recovers the standard duality relation at the level of the reduced quanton--detector state, while coherence associated with alternative causal orders remains inaccessible in this reduced description. To characterize this contribution, we introduce \emph{causal coherence}, defined as the coherence of the reduced order qubit in the causal-order basis. We further show that complementarity admits a state-dependent entropic formulation based on incompatible measurements on the causal degree of freedom, which arises from a universal uncertainty principle and provides a canonical operationally meaningful description. These results show that operational formulations of complementarity can depend on causal structure and cannot, in general, be fully characterized at the level of reduced quantum states alone.
The following two questions are examined: 1o Do the unobservable parameters possess precise, though unknown, values ? 2o If these unobservable values were known, would it be possible to make precise predictions of the reults of later measurements ? The answer is shown to be negative; the questions, therefore, are not meaningless, being capable of a falsification. The inquiry leads to the establishment of a principle of anomaly, more precisely speaking, of causal anomaly, which is to be added to Heisenberg's principle of indeterminacy. This principle states that the principle of action by contact is violated whenever definite values are assigned to the unobserved quantities, i. e., when an exhaustive interpretation of quantum mechanics is used. The two most important exhaustive interpretations are given by the corpuscle and the wave interpretation; each leads to causal anomalies, though for different places. The causal anomalies can be eliminated by the use of a restrictive interpretation, which separates statements about unobserved quantities, as a third propositional class, from true or false statements. Bohr and Heisenberg have called such statements meaningless, without being able to eliminate them completely; these statements arte thus merely shifted into the metalanguage. In another version of the restrictive interpretation such statements are assigned a third truth value, the value indeterminate, and quantum mechanics is then presented in the form of a three-valued logic. The inquiry is carried through without the presupposition of any philosophical conception; every interpretation is examined with respect to the consequences to which it leads. - H. R.'