The Noncommutative Values of Quantum Observables:
A New Perspective on Quantum Physics

Otto C.W. Kong (National Central University)

ABSTRACT

The article gives an introductory summary of our recent results from the perspective of the new conceptual notion of noncommutative values of quantum observables, where basic quantum mechanics is seen a theory of particle dynamics that is intrinsically noncommutative. Real numbers as an algebra is really an abstract mathematical model for the description of values of physical quantities. Quantum mechanics illustrates the limitation of that perspective. The noncommutative values preserve the algebraic relations among the observables in their values. Such a value contains the complete information the theory presents for an observable on a given state. All physical quantities, including coordinates of the physical space for describing the location of a particle take definite noncommutative values.

INTRODUCTION

Quantum mechanics has been deemed counter-intuitive. It has, however, been our deep belief that most, if not all, the difficulties in 'understanding' quantum mechanics may be results of our indulgence in using classical concepts to think about quantum physics. The classical perspectives are more familiar, but not necessarily more intuitive. The collection of real numbers as an abstract mathematical system, for example, is not at all intuitive. It's easy to forget that when we were students, we weren't able to have even a vague appreciation of what real numbers are without investing quite some effort. And after learning and using them for decades, most physicists are probably unable to give a clear definition of what they are. Moreover, most of the vague perspectives we have about them can actually be fulfilled by some other mathematical objects. Intuitions, indeed, are vague. Only their mathematical formulations are logically definite and rigorous. Modern mathematics has shown us that there can be different ways to formulate models incorporating our intuitions.

It is intuitive that an object has 'a' definite position in physical space. The latter being exactly the Newtonian model is not part of that intuition. Quantum mechanics certainly says a particle cannot have definite real number values for all its observables, including position observables, at any instant of time. It does not exclude the possibility of such definite values modeled with different mathematical objects. Furthermore, changing how the values of the position (coordinate) observables are to be described mathematically means changing the model of physical space. The resulting Heisenberg commutation relation is a key quantum feature. The ensuing uncertainty principle states the limitation of the best single real number descriptions of the values among the noncommuting observables. Getting from the algebra of physical observables to their values for a fixed state is, at least for classical physics, really a homomorphic map from the algebra of observables to the algebra of possible values. The algebra of real numbers is the good candidate for the latter only when the algebra of observables is commutative. Quantum mechanics asks for a noncommutative algebra to model the values. We have given, or rather identified, a concrete representation of the noncommutative values each as an infinite sequence of complex numbers [1, 2]; in fact, more than one such representation as variations on a single theme. A noncommutative value is a piece of quantum information which is equivalent to infinite pieces of classical, real number, information [3].

Readers may feel puzzled by our statement about the coordinates of the physical space as noncommutative quantities. The position observables in quantum mechanics, of course, commute among themselves. However, from the perspective of a quantum version of the (Galilean) relativity symmetry, it can be appreciated that the phase space (of a free particle) is the proper model for the idea of the physical space [4, 5]. It is exactly like Lorentz invariance dictates the model of Minkowski spacetime as the proper model for the notion within the theory. In the quantum regime, there is no invariant separation of the configuration/position space and momentum space. The phase as a whole is an irreducible representation. Actually, a pair of position and momentum observables are not, straightly speaking, independent coordinates. Their (noncommutative) values bear a fixed relationship given by the Heisenberg commutation relation. There is even a (noncommutative algebraic) notion of dimensionality which, in a way, says that the quantum phase space is three, rather than six, dimensional [1]. That phase space is indeed the familiar one as the (projective) Hilbert space of infinite, real-number geometric, dimensions. Quantum noncommutativity can be described on functions of the space as observables [6], and the set of position and momentum operators can be taken as an alternative coordinate system of it, which are noncommutative [7].

THE NONCOMMUTATIVE VALUES OF OBSERVABLES

Let us ask first what is the full information the theory of quantum mechanics gives an observable for a fixed state. The textbook answer is the full statistical distribution resulting for the corresponding projective measurements, and the correctness of this answer has been well verified. The distribution contains a large amount of information. Taking the eigenvalue answer from a single measurement gives little useful information about the property of the original state one intends to measure. The expectation value is the best single real number representation of the distribution. But we always have to measure the distribution and we can do that up to any required precision. Such a distribution can only be described by an infinite number of real numbers. The distributions do not serve as candidates for the values of the observables because we cannot combine them algebraically like getting the distribution of an observable as a sum or product of two observables from their distributions. The noncommutative value is otherwise quite similar to such a distribution, especially as seen from the current experimental technological point of view.

The starting point to understand the currently available concrete mathematical formulations of the noncommutative values is the identification of the observable algebra as one of functions on the (projective) Hilbert space as a symplectic manifold with a noncommutative product [6]. The rest can be seen as the natural logical consequence, given the right perspective. Take an orthonormal basis with n from 0 to ∞ of the Hilbert space, a quantum state can be written as where the complex number coordinates zⁿ are taken to satisfy the normalization condition . An observable given as the operator β has the expectation value function

Ref. [6] presents the 'Kähler product' between two such functions with the result being the expectation value function of the product operator. Explicitly [2, 7], we have

where is the standard Fubini-Study metric of the projective Hilbert space, space of rays hence individual physical states, as a Kähler manifold [8] here expressed in terms of the zⁿ as a set of homogeneous coordinates. For a fixed state, hence fixed zⁿ, the expectation value of βγ can be obtained from the values of f_β, f_γ and the values of the functions of their coordinate derivatives ∂_mf_β and ∂_n-f_γ. Then, we will need values for up to the n-th order derivatives to express the expectation value of βⁿ. This recall the idea of the Taylor series expansion as the description of the full information of a function f_β at a point, hence like the full information of β for the fixed state. We may have then, for each state, a homomorphic map from the observable algebra to the algebra elements of which are the sets of the coefficients of the local Taylor series for the operators. Each such set of coefficients can be seen as a noncommutative value with the noncommutative product between two to be retrieved from the Kähler product.

It happens that an algebraic isomorphism, for each fixed state, mapping each operator of the observable algebra to a sequence of infinite number of complex numbers, essentially the sets of the coefficients of the local Taylor series, has been presented in a 1996 Ph.D. dissertation [9], though the author did not seem to have the idea of that being the algebra of noncommutative values for the observables. It is also noticed that the higher order derivatives beyond the second order ones are not independent ones, hence can be neglected. The studies of Refs. [6, 9] give many important results on a full understanding of the symplectic geometric picture of quantum mechanics which dates back to the late 70s. A simple coordinate presentation of many of the results and more background references are given in Ref. [7]. The noncommutative value of Hermitian operator β can be written as

all to be taken as the complex number value at the fixed state. It can be easily checked that an eigenstate of an observable always has all corresponding  being zero, and degenerate eigenstates for an observable have identical noncommutative values. Moreover, β = cI gives f = c and all zero for the other parts of the noncommutative values on any state, hence only the zero operator has zero noncommutative values. The noncommutative product is then given by  with

where  is just the complex conjugate of .

PRACTICAL EXPERIMENTAL ISSUES

For readers who still need assurance that the mathematical description of a state in quantum mechanics is truly physical, we want to draw their attention to the practical experimental determination, or measurement, of a quantum state [10, 11]. Even more exciting is the recent success in monitoring a quantum state in real time including but not quite limited to the discontinuous 'jump' resulted from projective measurements [12]. With all that background, one can see that a noncommutative value, at least in terms of the complex numbers in the infinite sequence representation, can be experimentally determined, at least in principle. The expectation value can be determined. From quantum state tomography or otherwise, the state can be determined and hence its coordinate representation with a given basis can be obtained up to an overall phase factor. The matrix elements of the operator concerned can be determined. One may then obtain all complex number values of and from Eq. (3). Note that for a system that can be described effectively on a finite dimensional Hilbert space, we have only a finite number of and . For a single qubit (pure) state in particular, we have only two , which can be expressed with three real numbers, and a four real number full description of the three .

The notion of noncommutative value for an observable would not be of much experimental interest if its determination has to involve determining the state first. However, that does not have to be the case. Each noncommutative value is really an element of a noncommutative algebra, an abstract simple in itself. Its representation as a sequence of complex numbers is only one mathematical way to give it a more concrete realization for those familiar with real/complex numbers. Yet, the real numbers themselves as mathematical symbols are not really fundamentally any less abstract than elements of other algebras.

Let us look into the notion of a measurement of physical quantities very carefully. A measurement is a controlled interaction with the physical system we are interested in with a goal of extracting a particular information about its state before that interaction. It is actually not necessary to worry about the possible change of the state during the process. To actually obtain the information or the 'values' we are after may, however, be a complicated issue. In general, we need to have a good (theoretical) understanding of the physics of the measuring process and perform some calculations. The world is quantum and the information about a physical system is intrinsically quantum in nature, though it may be approximated by, or in many cases degraded to, the classical information represented by a few real numbers. Extracting a piece of quantum information from a system should really be taken as a kind of measuring process, and it is certainly not the one giving the 'real number answers'. For the direct measurements in which the answer is read out of an apparatus, the key is to have the right apparatus and a good calibration of the output scale. We essentially measure by comparison. We compare the 'value we measure, as indicated on the reading scale, with 'known value of the quantities which may be a conventionally chosen standard unit. The comparison itself never gives us the real number readings though. We put that in ourselves. The truth is that nothing in nature ever points to the idea of the physical quantities being real valued. In all (classical) measurements, it is our calibration of the output reading scale in the measuring device that enforces the reading as a real number, and even that can only be taken as an approximate range. The bottom line is that the real numbers as the representation of the values of physical quantities are nothing more than a mathematical model we use to describe nature. But the model shows inadequacy in the quantum regime. To consider the full practical implementations of determining the noncommutative value more directly, one may have to devise an appropriate apparatus and calibrate its output with the noncommutative values. It is essentially about describing quantum information directly as quantum information. With advancement in the latter technology and some ingenious experimental ideas, we may achieve that in the future.

NONCOMMUTATIVE PHYSICS FROM A GROUP THEORETICAL PERSPECTIVE

Our studies on the noncommutative values of quantum observables is a complementary side of our approach to the physics of noncommutative geometry [13]. We see quantum mechanics as about a noncommutative geometric model of the physical space [5, 7], and as such the simplest practical version of it. The geometry of quantum mechanics is probably the noncommutative analog of the Euclidean space of the commutative case. Deep microscopic physics may involve more general noncommutativity as one among the position operators and among the momentum operators [14, 15]. A dynamical noncommutativity may be what true quantum gravity is about. In relation to that, it is interesting to note that the projective Hilbert space is a curved one with a curvature essentially fixed by the noncommutativity [6]. Physicists first involvement with noncommutativity is probably from group theory, namely the noncommutativity of rotations as symmetry of the Newtonian space. Lie group mathematics actually goes very far. The basic quantum noncommutativity from Heisenberg gives a natural picture of quantum mechanics as essentially an irreducible representation of the Heisenberg-Weyl symmetry. Such representation theory properly extended gives essentially the full theory and its approximate versions as contraction limits [5]. We have put together under a single framework the group theoretical formulation of a Lorentz invariant quantum mechanics and all its approximations as the 'nonrelativistic' and/or classical limits [16]. The theory has a notion of noncommutative Minkowski spacetime and a Minkowski metric operator on the corresponding Hilbert space. Going up to construct the fully noncommutative theory [14, 15] to which they are approximations of is a window the framework offers.

Final Remarks

Most physicists agree with Bohr that quantum mechanics is a complete and very successful theory. It can actually be complete even in the demanding sense Einstein asked for, provided that we accept the notion of the noncommutative values as the definite values of the physical quantities. They are fully predicted by the theory, and, though only quite indirectly with known experimental techniques, verifiable. No probability has to be involved. No hidden variables are required. It is the known quantum variables that have some kind of hidden part of the values. Our old insistence on admitting only real numbers as values of an observable can hide the full information that quantum world contains as correctly predicted by the theory. The noncommutative values, at least, give good theoretical way to think about the theory and theories beyond.

Acknowledgments
This work has been supported by the research grant number 109-2112-M-008-016 of the MOST of Taiwan.

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