_{1}

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We discuss an alternative version of non-relativistic Newtonian mechanics in terms of real Hilbert space mathematical framework. It is demonstrated that the physics of this scheme is in accordance with the standard formulation. Heisenberg-Schrodinger non-relativistic quantum mechanics is considered adequate and complete. Since the suggested theory is dispersion free, linear superposition principle is not violated but cannot affect results of measurements due to spectral decomposition theorem for self-adjoint operators (the collapse of wave function).

The debates about the interconnection between the hidden laws of nature and our ability to extract the information necessary to formulate them have perhaps a history as long as study of physics itself. The content of our paper is not related to the philosophical or metaphysical aspects of those discussions. Instead, we choose certain point of view without intention to defend it or to convince the reader that it is the only possible approach. We simply present how the process of knowledge acquisition is realized within that approach. We explore the analogy with the structure of field theories (classical electrodynamics, general relativity and non-relativistic quantum mechanics) and make distinction between the unobservable kinematical quantities which characterize the physical system and the measurable variables which define its dynamics. Since the main distinction between the classical and the quantum physics is in presence of new kinematical quantities―phases, one should learn how to measure the corresponding phase differences. We demonstrate that the required measurement may be performed using special experimental arrangement which we call the quantum reference frames. The use of these reference frames allows communicating the hidden unobservable information into the instruments of the observer. Simultaneously it explains why the elementary unit of the communication is given in terms of indivisible bit.

In order to achieve our goal we establish for classical mechanics the structural framework similar to the one used in quantum theory. We restrict ourselves to description of single particle states and prefer here to avoid complications introduced by special relativity. In order to make clear the mathematical and correspondent physical content for successive discussion, I will quote the following statement [

If

with

This theorem appears several times [

Now, let us consider famous E. Schrödinger cat example [

1) The cat may be presented as a quantum mechanical system and not as a classical measurement instrument;

2) The system state is described by the following linear superposition of pure states:

E. Schrödinger did not continue discussion after that point. But since a cat is in the superposition state, this will lead to the spread of wave packet within time uncertainty predicted by W. Heisenberg. The curious experimenter would find cat “blurred” over entire volume of the chamber and disappeared (from classical point of view) together with his smile (notice that if it was correct, then the quantum mechanics would provide proper unification between L. Carroll and E. Schrödinger fantasies). It was remarkable that E. Schrödinger concluded discussion of cat paradox by the following statement:

“It is typical of these cases that an indeterminacy originally restricted to the atomic domain becomes transformed into macroscopic indeterminacy, which can then be resolved by direct observation”.

That statement is in contradiction with the J. von Neumann conjecture [

First of all it should be defined what we mean by real Hilbert space. We should maintain the connection with quantum theory as close as possible and assure the proper extension to relativistic version.

In addition, a scheme should incorporate classical electrodynamics through application of principle of local gauge invariance. Therefore we will use the following definitions:

^{(*)}Scalar product in this framework is defined by:

with underlined numerical basis of dimension two (complex numbers). This implies that

In particular,

Notice that in quantum theory the relevant scalar products associated with observable quantities are always real. Since in classical mechanics every dynamical variable is observable, we will further discuss only self-adjoint operators. They satisfy the following algebra:

1)

This statement will be justified after introduction of a product of two self-adjoint operators.

2)

Proof:

Obviously,

Then

Therefore

with respect to real scalar product defined above.

Since the product of any pair of self-adjoint operators is a self-adjoint operator, the product of an arbitrary number of self-adjoint operators is a self-adjoint operator.

Now let us examine properties of self-adjoint operators in these schemes.

1) Spectral Decomposition Theorem: For every self-adjoint (hermitian) operator

The components of this basis are the solutions of the equation

which is a consequence of Equation (1).

The careful reader may verify line by line that there is no difference between complex and real Hilbert spaces as defined above with respect to spectral decomposition theorem. There exists vast literature on the topic but the books of R. Courant and D. Hilbert [

Here perhaps I should add the important remark. When we write

we usually say that it provides physical value(s) of the observable

provides the values of the observable in quantum theory. Therefore, an additional relation is required in order to associate them with the results of measurements.

2) The necessary and sufficient condition for two or any number of hermitian operators to have a common set of eigenfuctions which form a complete orthonormal basis in Hilbert space is that they are mutually commuted. Since the product of two or any number of mutually commuting hermitian operators is again a hermitian operator (and commutes with each of its components), it has the same set of eigenfunctions. Indeed, every one of them in that basis is dispersion free.

Hence, the real Hilbert space as defined above provides realization of dispersion free physical theory.

Moreover, since the coordinate

3) Another mathematical statement that may have interesting physical realization (we will discuss it later) is valid in real Hilbert space: for an arbitrary set of mutually commuting hermitian operators

Now it become manifestly obvious that real Hilbert space provides a convenient arena for Newtonian mechanics. Finally, let us demonstrate that the equation

is equivalent to the Heisenberg quantization condition.

It is well known that the classical equations of motion

have the following form in terms of hermitian operators [

if we choose

and

then using Equation (3) we obtain

which are the quantum equations of motion written in the Heisenberg representation.

The Equation (3) is the fundamental relation that defines the results of measurements. For the model example of a particle in an infinite spherical well, only discrete solutions of right hand side embedded into continuous spectrum of left hand side will be revealed.

In the previous section the formulation of Newtonian dynamics was achieved and was even demonstrated that the same dynamical law still governs time evolution of the system in quantum physics. It was discovered by E. Schrödinger [

We may say that for the mathematical structure it is sufficient to be legal and legitimate if its foundation is based on mutually consistent set of assumptions. It is not sufficient for physics: physics is an empirical science. The realization of that requirement is performed through introduction of properly defined measurement instruments and procedures. The structure of Newtonian mechanics represents the basic ingredients needed to achieve that. In addition to the formulation of the time evolution of the system (dynamics), the equations of motion should earn status of physical law. The latter requirement is satisfied by introduction of the reference frames and rules as how they are connected with each other. It is meaningless to discuss any law or relation without its universality with respect to a chosen and well defined infinite set of reference frames. In the classical mechanics, it turns out that the definition of the suitable reference frame (inertial systems) occurs through the idealization of the free moving body isolated from its environment. Then the connection between those frames is given in terms of the motion of such a body. This is the content of the first law of the Newtonian mechanics (Galileo law of inertia), which is indeed consistent with the fundamental equations of motion. We associate that body, located at the origin of a given reference frame, with an appropriate set of measurement instruments. Therefore the origin of the reference frame should be defined with certainty and the measurement instruments should obey laws of classical physics.

In the scheme suggested here, both requirements are met and Equation (3) assures that all relevant information about properties of investigated quantum mechanical system is available. Equation (3) plays a role analogous to the third Newton law.

A single isolated sample of the experimental data has no meaning in classical physics. Only repetitions of the sample will confirm that the obtained result represents the objective reality. The requirement that the system state remains unchanged during the experiment was never fulfilled; even the system invariants (for example in collisions) may change. What is essential is that if the consecutive (in time) measurements on the same system are not legal, the repetition of the measurement should be assured by possibility to prepare a system identical to the original one. As pointed out by E. Schrödinger [

The collection of the obtained results is now the subject of the standard techniques for the data processing. In case the system under test is a classically defined material point (coordinate and momentum are mutually commuted hermitian operators), one will obtain a picture sharply concentrated around a single isolated point. In case when the system obeys the laws of quantum physics, one will get a picture of a spatially extended object; the number of required samples is determined by the classical methods of image and/or signal processing [

The notion of eigenschaften operator was introduced by J. von Neumann [

Together with the requirement of being observable (

Theorem.

If

Then

Proof:

1) Suppose

then

2) Suppose

then

From

Let us consider first the two-dimensional case. From the Equation (5)

and due to the spectral decomposition theorem, we have

Since

finally we obtain

Now in terms of matrix mechanics we have

with

Then

Since here we discuss the measurements of relevant parameters of the quantum mechanical systems with non-vanishing dispersion, we will consider only

or

The matrix elements

and

are all we need to know about the quantum state. Both are measurable,

Then dropping the overall phase factor, we obtain

Using the relations (12) we obtain the following most general solution

In particular, for

well known in image processing applications.

The remarkable feature of the measurement process is that the measurement devices are macroscopic, obeying the laws of classical physics, whereas the systems under test belong to the microscopic world and behave quantum mechanically. Indeed, the measurement setup should assure that the obtained results represent the objective properties of the investigated physical system and not a free subjective imagination of the observer. That task is performed in classical physics by introduction of reference frames such that the detector location defines the frame origin and the set of auxiliary macroscopic devices allows establishing the connection and the communication between the frames separated by the finite space-time interval (the comparison of the obtained empirical data must always be performed by the same observer). Similarly, in order to perform the measurement of the relevant quantum dynamical variable one should include in the classical setup a set of auxiliary macroscopic devices which produce the necessary beam-splitting. Then the required phase differences are measured in the usual way. This setup and recording procedure may be viewed as the general holographic detection. But in contrast to the classical physics, measurements in quantum mechanics should provide simultaneous information about all relevant relative amplitudes (pure states and the transitions between them) and all relevant relative phases. Simultaneity is needed since in general the measurement changes the state of the system (in quantum physics and in classical physics as well). We illustrate that by explicit examples and describe general features of the corresponding experimental setup which we identify as the quantum reference frame.

We address the following questions:

1) What is the difference between “on-off” and “or-and” switches in terms of quantum mechanical self-adjoint operators (observables)?

2) How the transition amplitudes between the stationary (pure) states are incorporated naturally and symmetrically together with the amplitudes of these states?^{ }

3) Is it possible to measure

4) If it is possible, may that measurement be performed using only macroscopic devices?

It is well known [

Let us expose the content of our discussion viewed from the eyes of Schrödinger’s cat totally confused by endless debates about his destiny. The usual justification of apparent uncertainty in it refers to HDR. But the empirical evidence clearly tells us that the initial assumption that the cat may be considered as quantum mechanical system containing an inherent indeterminacy which “becomes transformed into macroscopic indeterminacy” [

The lossless beam splitter here is the macroscopic device which participates actively in the detection procedure (

In contrast, in microscopic quantum mechanical world (quantum optics) we are required to measure also the phase differences in order to obtain all existent and necessary information about the original object. This may be done using similar setup, for example (

It was demonstrated that non-relativistic classical mechanics might be reformulated in terms of real Hilbert space, with an underlined numerical system of complex numbers. It is worthwhile to mention that similar structures based on real quaternions and real octonions exist. The presence of rich phase structure in the definition of wave functions (system states) should allow the axiomatic introduction of electromagnetic and gravitational interactions by means of application of the principle of the local gauge invariance. The relativistic version of the theory is expected to emerge naturally in suggested frameworks. The present paper is devoted mainly to the problems related to the measurement theory. Within a classical world we are working in Heisenberg representation. Hilbert space appears to be uniquely defined and rigid and plays a role of passive arena for the events associated with the dynamics of the physical system. The space-time continuum plays a similar role in the standard formulation of Newtonian mechanics. It seems reasonable to expect that these arenas are actually identical and the J. von Neumann theorem mentioned above (statement 3) may assign to it the dynamical content.

Daniel Sepunaru, (2015) Elements of Real Hilbert Spaces Theory. Open Access Library Journal,02,1-9. doi: 10.4236/oalib.1101554