The Noncommutative Values of Quantum Observables

TL;DR

This paper explores the concept of noncommutative values for quantum observables, proposing a new way to represent their values beyond classical real numbers, using infinite sets of complex numbers.

Contribution

It introduces the idea of noncommutative values for quantum observables and discusses their theoretical and practical feasibility.

Findings

Noncommutative values can be represented as infinite sets of complex numbers.

The proposed representation makes sense both theoretically and practically.

Challenges classical notions of observable values in quantum mechanics.

Abstract

We discuss the notion about physical quantities as having values represented by real numbers, and its limiting to describe nature to be understood in relation to our appreciation that the quantum theory is a better theory of natural phenomena than its classical analog. Getting from the algebra of physical observables to their values on a fixed state is, at least for classical physics, really a homomorphic map from the algebra into the real number algebra. The limitation of the latter to represent the values of quantum observables with noncommutating algebraic relation is obvious. We introduce and discuss the idea of the noncommutative values of quantum observables and its feasibility, arguing that at least in terms of the representation of such a value as an infinite set of complex number, the idea makes reasonable sense theoretically as well as practically.

Equations

$$H_{\beta }(z^n,{\bar{z}}^n)=\frac{1}{2\hslash }⟨\varphi |\beta |\varphi ⟩=\frac{1}{2\hslash }\sum {\bar{z}}^m{z}^n⟨m|\beta |n⟩,$$

$$f_{\beta }(z^n,{\bar{z}}^n)=\frac{2\hslash }{|z{|}^2}{H}_{\beta }(z^n,{\bar{z}}^n)=\frac{1}{|z{|}^2}\sum {\bar{z}}^m{z}^n⟨m|\beta |n⟩,$$

$$H_{\beta }{\star }_K{H}_{\gamma }=\hslash {\partial }_m{H}_{\beta }{G}^{m\bar{n}}{\partial }_{\bar{n}}{H}_{\gamma },$$

$$f_{\beta }{\star }_{\kappa }{f}_{\gamma }=f_{\beta }{f}_{\gamma }+\hslash {\partial }_m{f}_{\beta }{\tilde{g}}^{m\bar{n}}{\partial }_{\bar{n}}{f}_{\gamma },$$

$$H_{\beta \gamma }=\hslash {\tilde{X}}_{{\beta }_m}{G}^{m\bar{n}}{\tilde{X}}_{{\gamma }_{\bar{n}}}=2\hslash \sum _n{\tilde{X}}_{{\beta }_n}{\tilde{X}}_{{\gamma }_{\bar{n}}},$$

$${\tilde{X}}_{\beta {\gamma }_n}=2i\hslash \sum _m{\tilde{X}}_{{\beta }_m}{\tilde{K}}_{{\gamma }_{n\bar{m}}},$$

$${\tilde{X}}_{\beta {\gamma }_{\bar{n}}}=2i\hslash \sum _m{\tilde{K}}_{{\beta }_{m\bar{n}}}{\tilde{X}}_{{\gamma }_{\bar{m}}},$$

$${\tilde{K}}_{\beta {\gamma }_{m\bar{n}}}=2i\hslash \sum _l{\tilde{K}}_{{\beta }_{l\bar{n}}}{\tilde{K}}_{{\gamma }_{m\bar{l}}}.$$

$$\displaystyle{\tilde{X}}_{\!\scriptscriptstyle\beta_{n}}=i\partial_{n}f_{\!\scriptscriptstyle\beta}=\frac{-i}{|z|^{2}}\Big{[}f_{\!\scriptscriptstyle\beta}\bar{z}_{n}-\sum_{m}\bar{z}^{m}\left\langle m|\beta|n\right\rangle\Big{]}\;,$$

$$\displaystyle{\tilde{X}}_{\!\scriptscriptstyle\beta\bar{n}}=-i\partial_{\bar{n}}f_{\!\scriptscriptstyle\beta}=\frac{i}{|z|^{2}}\Big{[}f_{\!\scriptscriptstyle\beta}{z}_{n}-\sum_{m}{z}^{m}\left\langle n|\beta|m\right\rangle\Big{]}\;,$$

$${\tilde{K}}_{{\beta }_{m\bar{n}}}=-{\tilde{K}}_{{\beta }_{\bar{n}m}}=-i{\partial }_m{\partial }_{\bar{n}}{f}_{\beta }$$

$$\displaystyle\qquad=\frac{i}{|z|^{2}}\big{[}f_{\!\scriptscriptstyle\beta}\,\delta_{m\bar{n}}+i{\bar{z}_{m}}{\tilde{X}}_{\!\scriptscriptstyle\beta_{\bar{n}}}-i{{z}_{n}}{\tilde{X}}_{\!\scriptscriptstyle\beta_{m}}-\left\langle n|\beta|m\right\rangle\big{]}\;.$$

$$\displaystyle{{X}}_{\!\scriptscriptstyle\beta_{n}}=i\partial_{n}f_{\!\scriptscriptstyle\beta}=\frac{-i}{(1+|w|^{2})}\Big{[}f_{\!\scriptscriptstyle\beta}\bar{w}_{n}-\sum_{m}\bar{w}^{m}\left\langle m|\beta|n\right\rangle\Big{]}\;,$$

$$\displaystyle{{X}}_{\!\scriptscriptstyle\beta\bar{n}}=-i\partial_{\bar{n}}f_{\!\scriptscriptstyle\beta}=\frac{i}{(1+|w|^{2})}\Big{[}f_{\!\scriptscriptstyle\beta}{w_{n}}-\sum_{m}{w}^{m}\left\langle n|\beta|m\right\rangle\Big{]}\;,$$

$$K_{{\beta }_{m\bar{n}}}=-K_{{\beta }_{\bar{n}m}}=-i{\partial }_m{\partial }_{\bar{n}}{f}_{\beta }$$

$$\displaystyle\qquad=\frac{i}{(1+|w|^{2})}\big{[}f_{\!\scriptscriptstyle\beta}\,\delta_{m\bar{n}}+i{\bar{w}_{m}}{{X}}_{\!\scriptscriptstyle\beta_{\bar{n}}}-i{w}_{n}{{X}}_{\!\scriptscriptstyle\beta_{m}}-\left\langle n|\beta|m\right\rangle\big{]}\;,$$

$$f_{\beta \gamma }=f_{\beta }{f}_{\gamma }+|z{|}^2\sum _n{\tilde{X}}_{{\beta }_n}{\tilde{X}}_{{\gamma }_{\bar{n}}},$$

$${\tilde{X}}_{\beta {\gamma }_n}=f_{\beta }{\tilde{X}}_{{\gamma }_n}+{\tilde{X}}_{{\beta }_n}{f}_{\gamma }+i|z{|}^2\sum _m{\tilde{X}}_{{\beta }_m}{\tilde{K}}_{{\gamma }_{n\bar{m}}},$$

$${\tilde{X}}_{\beta {\gamma }_{\bar{n}}}=f_{\beta }{\tilde{X}}_{{\gamma }_{\bar{n}}}+{\tilde{X}}_{{\beta }_{\bar{n}}}{f}_{\gamma }+i|z{|}^2\sum _m{\tilde{K}}_{{\beta }_{m\bar{n}}}{\tilde{X}}_{{\gamma }_{\bar{m}}},$$

$${\tilde{K}}_{\beta {\gamma }_{m\bar{n}}}=f_{\beta }{\tilde{K}}_{{\gamma }_{m\bar{n}}}+{\tilde{K}}_{{\beta }_{m\bar{n}}}{f}_{\gamma }+i|z{|}^2\sum _l{\tilde{K}}_{{\beta }_{l\bar{n}}}{\tilde{K}}_{{\gamma }_{m\bar{l}}}-i{\tilde{X}}_{{\beta }_{\bar{n}}}{\tilde{X}}_{{\gamma }_m}+\frac{i|z{|}^2{\tilde{g}}_{m\bar{n}}}{\hslash }\sum _l{\tilde{X}}_{{\beta }_l}{\tilde{X}}_{{\gamma }_{\bar{l}}},$$

$$f_{\beta \gamma }=f_{\beta }{f}_{\gamma }+\hslash X_{{\beta }_m}{g}^{m\bar{n}}{X}_{{\gamma }_{\bar{n}}},$$

$$X_{\beta {\gamma }_n}=f_{\beta }{X}_{{\gamma }_n}+X_{{\beta }_n}{f}_{\gamma }+i\hslash X_{{\beta }_l}{g}^{l\bar{m}}{K}_{{\gamma }_{n\bar{m}}},$$

$$X_{\beta {\gamma }_{\bar{n}}}=f_{\beta }{X}_{{\gamma }_{\bar{n}}}+X_{{\beta }_{\bar{n}}}{f}_{\gamma }+i\hslash K_{{\beta }_{l\bar{n}}}{g}^{l\bar{m}}{X}_{{\gamma }_{\bar{m}}},$$

$$K_{\beta {\gamma }_{m\bar{n}}}=f_{\beta }{K}_{{\gamma }_{m\bar{n}}}+K_{{\beta }_{m\bar{n}}}{f}_{\gamma }+i\hslash K_{{\beta }_{l\bar{n}}}{g}^{l\bar{o}}{K}_{{\gamma }_{m\bar{o}}}-i{X}_{{\beta }_{\bar{n}}}{\tilde{X}}_{{\gamma }_m}+i{g}_{m\bar{n}}{g}^{l\bar{o}}{X}_{\beta l}{X}_{\gamma \bar{o}}.$$

Full text

1 Introduction

Newton presented his theory of mechanics with the hypothetical notion of a point particle with an unambiguous position (the point) in the physical space of assumed three dimensional Euclidean geometry as essentially a product of the real number lines. Note that kind of geometry was the only mathematical model for the space as a continuum in the mathematics of Newton’s time. The three independent coordinates of a particle then give the basic observables the values of which are given by the real numbers. Classical physics is then a real number physics. Physical quantities or observables, of the particle in the case, are modeled by the real valued variables. However, in order to allow the theory to work in all inertial frames, instead of only in an absolute frame of reference, with respect to which we measure all the motion, the basic independent variables have to be extended to include the velocity or momentum components. We have the full set of six real coordinates of a point in the phase space giving a unique state of the particle. The phase space is then also essentially modeled by the Euclidean geometry, though the idea of a metric or distance between two states is not considered to be of any use. The picture of a particle as occupying a fixed position in physical space is an intuitive one, a desirable if not an absolutely necessary feature of the theory of particle dynamics (versus field theories) that we have been told we have to give up in quantum mechanics. A key feature of our study of quantum spacetime is to restore that. A quantum particle sure cannot have a fixed position in the Newtonian model of the physical space. That may rather be taken as an indication that the latter model fails the intuitive notion of a model for the physical space and demands efforts to find a good enough model that works. In fact, the only physical notion of the physical space in a theory of particle dynamics is the collection of all possible positions of a free particle. Actually, the model for the physical space is such an integral part of the theory that changing the theory assuming the model to be unchanged may not be really sensible.

The most important point of departure from classical physics in quantum mechanics is the realization that quantum observables, the necessary model description of real physical observables as seen at high enough precision beyond the classical domain, in general do not commute. In particular, the conjugate pairs of position and momentum observables each is an non-commuting pair. The position observables, however, commute among themselves. The latter might be the reason why most physicists believe it is still fine to keep the Newtonian model for the physical space. But then the quantum phase space has been realized to be very different from the Cartesian product of the three dimensional Euclidean spaces for the position and the momentum. It is an infinite dimensional Hilbert space $\mathcal{H}$, or better taken as the projective Hilbert space $\mathcal{P}$ (or a $U(1)$ principal bundle of it). With time came the appreciation of the latter as a symplectic manifold with the Schrödinger equation giving an infinite number of pairs of Hamilton equation of motion on it (see our companion paper [1] and references therein). That begs the questions why we cannot take a certain Lagrangian submanifold of it, which should correspond to the configuration space for a free particle, as a model for the physical space, and how to reconcile the fact of there being only the three pairs of position and momentum observables with the infinite dimensional phase space which could be taken as having an infinite number of pairs of position and momentum coordinates. Answer to the first question has been presented in Refs.[2, 3]. Those position and momentum coordinates are like the space and time coordinates of the Minkowski spacetime, separable only in the Newtonian approximation. In the quantum setting, there cannot be an independent notion of a configuration space, rather it can only be seen as part of the phase space. The phase space should be taken as the proper model for a sensible notion of something like the configuration space, hence (that for a free particle) should be taken as the model for the physical space. The current study, together with Ref.[1], is to address the second question. The former paper looks into how to think about the noncommutative geometric picture of the otherwise well known infinite dimensional Kähler manifold, to the point of illustrating a quite explicit description of a transformation between the infinite number of complex number coordinates to the six noncommutative coordinates as complex combinations of the pairs of position and momentum operators. Matching of the full differential and symplectic structures are presented in that language. The current article focus on the new notion of noncommutative value for a (quantum) observable, mostly through a description of such a value in terms of an infinite set of real numbers. The results establish the perspective that the information content of the set of the six noncommutative coordinates is the same as the of the infinite set of real/complex number coordinates. We take it as a pure theoretical/mathematical question first, and discuss the issues of the practical implementation of the results only at the end. A complete real number coordinate summary version of the key results here using simple mathematics, by taking the Hilbert space as phase space, with emphasis on the integrated perspectives for an intuitive picture of quantum/noncommutative physics is presented in Ref.[4] for a more general readership.

2 The Noncommutative Values of Observables

Let us focus first on the observables. Like the classical case, we think about all quantum observables as functions of the basic set of independent observables, the position and momentum ones. Each such classical observable is then a real variable in itself, while in the quantum case it is an element of a noncommutative observable algebra which has been modeled by the operators on $\mathcal{H}$. What about the values for such operators on a state? In the classical case, the value of an observable $a(p^{i},x^{i})$ for a state $(p^{i}_{o},x^{i}_{o})$ is the real number $a(p^{i}_{o},x^{i}_{o})$. Considering the values for all observables in the algebra at the same time, we have an evaluation map for each state that maps all the variables $a(p^{i},x^{i})$ to the real number values of $a(p^{i}_{o},x^{i}_{o})$. Such a map has to be a homomorphism between the observable algebra and the real numbers as an algebra. That is to say the algebraic relationship with a number of observables has to be respected by their values for any state. The evaluation map given by taking the real number values of the observables as functions of the states is obviously such a homomorphism. To push that reasoning to the algebra of quantum observables, we need a picture of the values for $\hat{x}^{i}$ and $\hat{p}^{i}$ satisfying also the commutation relation $[\hat{x}^{i},\hat{p}^{j}]=i\hbar\delta^{ij}$ as the image of an evaluation map for a state here denoted by $[\phi]$. $\left|\phi\right\rangle$ generally denotes a state vector in the Hilbert space. An actually physical state of course corresponds to the ray determined by any such vector related to one another by a constant multiple. Each physical state would be seen as essentially an evaluation map, as in the classical case, which fixes the values of all observables, the values they have for the state. However, the algebra obtained as the image of such a map has then to be a noncommutative algebra, for which a subalgebra of the real number algebra certainly cannot do. That in itself does not necessarily imply that we cannot have such maps, thus to have the fixed values for the quantum observables for a state. We cannot have the fixed real number values. So long as we are willing to consider using some noncommutative algebraic system to represent those values, it is plausible to have the definite values. The key purpose of this article is to give an explicit description of the kind of algebraic model, one we may call an algebra of noncommutative numbers. Actually, an isomorphic description of the observable algebra with reference to a definite state has been presented in a 1996 Ph.D. dissertation [5], though the author(s) did not seem to have the idea of that being the algebra of noncommutative (number) values for the observables. What the author(s) called *the algebra of symmetry data, with a particular set of infinite number of complex numbers as an element, is really the candidate for the algebra of noncommutative values, may be called the noncommutative number values, of the observables in a quantum theory. That is the key proposition in this article. * We aim at presenting below a more direct picture of that algebraic story and start to seek a better understanding and depiction of physics in terms of such noncommutative number values of the observables.

To prepare for the appreciation of the mathematics, we first want to note that an operator has really the information content of an infinite number of real/complex numbers. Taking an orthonormal basis for $\mathcal{H}$, for example the eigenstates of the ‘three dimensional’ harmonic oscillator, an operator is completely characterized by the infinite set of matrix elements. However, the picture is independent of any state. The symmetry data on a state for each noncommutative observable/variable is exactly a state specific set of infinite number of complex numbers characterizing the observable hence giving the noncommutative value of it, and such sets for the various observables have a noncommutative product that gives the set for the value of the product observable from the values, i.e. the two sets of complex numbers, for the two observables composing it. Of course such a set of complex numbers is only one mathematical way to represent the noncommutative value as an element of a noncommutative algebra. It is the way that gives it a more concrete realization for those familiar with the real/complex number values. Yet, the real numbers themselves as mathematical symbols are not really fundamentally any less abstract than elements of other algebras.

To sketch the story of the symmetry data in Ref.[5] and present our formulation, we have to first introduce two isomorphic descriptions of the observable algebra, which are more or less given in the reference, but actually have their origin in a very important earlier paper by Cirelli, Manià, and Pizzocchero [6] to which we owe much for our current understanding of the geometric structures behind quantum mechanics. Ref.[5] takes it quite a bit further, clarifying the geometric relations between $\mathcal{H}$ and $\mathcal{P}$ from the perspective of quantum mechanics and presenting the so-called symmetry data as a state dependent algebra isomorphic to the observable algebra. Apart from advancing the idea of the latter as candidates for the noncommutative values of observables, we will also present more explicit results in terms of standard complex coordinates to give readers easy comprehension of the key mathematics. Some details are left to Ref.[1] though. The algebraic isomorphisms are $H:\beta\to H_{\!\scriptscriptstyle\beta}(z^{n},\bar{z}^{n})$ and $f:\beta\to f_{\!\scriptscriptstyle\beta}(z^{n},\bar{z}^{n})$ with

[TABLE]

and

[TABLE]

for an operator $\beta$, of which we prefer to think about as a function of the coordinate operators $\beta(\hat{p}^{i},\hat{x}^{i})$, with state vector $\left|\phi\right\rangle=\sum z^{n}\left|n\right\rangle$; $\left|n\right\rangle$ with $n$ from 0 to $\infty$ denote an orthonormal basis of $\mathcal{H}$ and $z^{n}$ (with the index $n$) the complex coordinates of a point in $\mathcal{H}$ as a Kähler manifold; $|z|$ is then the norm of the state vector, or $|z|^{2}=z_{n}\bar{z}^{n}$, Einstein summation assumed as throughout the paper. $f_{\!\scriptscriptstyle\beta}$ can be taken as functions on $\mathcal{H}$ or functions on $\mathcal{P}$ here in homogeneous coordinates. Note that they are independent of the normalization of $\left|\phi\right\rangle$. The algebras of $H_{\!\scriptscriptstyle\beta}$ or $f_{\!\scriptscriptstyle\beta}$ functions of course each need to have a noncommutative product to match that of the operator product. They are the Kähler products given by

[TABLE]

and

[TABLE]

where $G^{{m}\bar{{n}}}=2\delta^{m\bar{n}}$ is the inverse metric of $\mathcal{H}$ and $\tilde{g}^{m\bar{n}}$ that of $\mathcal{P}$. The metric of $\mathcal{P}$ is the standard Fubini-Study metric. When expressed in terms of the homogeneous coordinates, the latter is degenerate. However, we can still have $\tilde{g}^{m\bar{n}}$ from Killing reduction of $\mathcal{H}-\{0\}$ [5, 1], given as $\frac{1}{\hbar}\left(|z|^{2}\delta^{m\bar{n}}-{z}^{{m}}\bar{z}^{{n}}\right)$ which can be used in the Kähler product calculation. We can also think about $\mathcal{H}-\{0\}$ as a fiber bundle with $\mathcal{P}$ as the base manifold and the degenerate metric independent of the fiber coordinates. The Kähler product may otherwise be given for the $f_{\!\scriptscriptstyle\beta}$ functions expressed in terms of a set of affine coordinates, for example $w^{n}=\frac{z^{n}}{z^{\scriptscriptstyle 0}}$ for $n\neq 0$ with the non-degenerate Fubini-Study metric [1]. In that case, we have $f_{\!\beta}\star_{\kappa}f_{\!\gamma}=f_{\!\beta}f_{\!\gamma}+\hbar\,\partial_{m}f_{\!\beta}\,{g}^{m\bar{n}}\partial_{\bar{n}}f_{\!\gamma}$, with ${g}^{m\bar{n}}=\frac{1}{\hbar}({1+|w|^{2}})\left(\delta^{m\bar{n}}+{w^{m}\bar{w}^{n}}\right)$. We have $H_{\!\scriptscriptstyle\beta\gamma}=H_{\!\scriptscriptstyle\beta}\star_{\!\scriptscriptstyle K}H_{\!\scriptscriptstyle\gamma}$ and $f_{\!\scriptscriptstyle\beta\gamma}=f_{\!\scriptscriptstyle\beta}\star_{\kappa}f_{\!\scriptscriptstyle\gamma}$ which can be easily verified explicitly in terms of generic matrix elements for the operators based on the coordinate.

The symmetry data for an operator $\beta$ at a point $[\phi]$ on $\mathcal{P}$ is given in Ref.[5] as a triplet $(f_{\!\scriptscriptstyle\beta},X_{\!\scriptscriptstyle\beta},K_{\!\scriptscriptstyle\beta})$ with $X_{\!\scriptscriptstyle\beta}$ being a covector of the Hamiltonian vector field for $f_{\!\scriptscriptstyle\beta}$, and $K_{\!\scriptscriptstyle\beta}$ a 2-form as the covariant derivative of $X_{\!\scriptscriptstyle\beta}$, all taken with values at $[\phi]$. Explicit expressions for the symmetry data for the Poisson bracket and Riemann bracket of two $f_{\!\beta}$ functions, corresponding to commutator and anticommutator of the operators [1, 5, 6], in terms of the two sets of $(f_{\!\scriptscriptstyle\beta},X_{\!\scriptscriptstyle\beta},K_{\!\scriptscriptstyle\beta})$ for the operators, are presented [5]. We present here below our reformulation of the picture which we believe makes more transparent all otherwise hidden in the abstract mathematics. And we give directly the product of the algebra of symmetry data at $[\phi]$ in terms of the Kähler product for the original $f_{\!\scriptscriptstyle\beta}$ functions. The Kähler product among the $f_{\!\scriptscriptstyle\beta}$ functions represents the operator product, to which the Poisson and Riemann brackets are simply the antisymmetric and symmetric parts.

As usual, mathematics behind quantum mechanics is easier to present on $\mathcal{H}$. Let us first look at the symmetry data for the $H_{\!\scriptscriptstyle\beta}$ functions. For each operator $\beta$, it is given as $(H_{\!\scriptscriptstyle\beta},\widetilde{X}_{\!\scriptscriptstyle\beta_{n}},\widetilde{X}_{\!\scriptscriptstyle\beta_{\bar{n}}},\widetilde{K}_{\!\scriptscriptstyle\beta_{m\bar{n}}}\!\!=\!{\nabla}_{\!\!{{m}}}\widetilde{X}_{\!\scriptscriptstyle\beta_{\bar{n}}}$), where $\widetilde{X}_{\!\scriptscriptstyle\beta}$ is the Hamiltonian vector field of $H_{\!\scriptscriptstyle\beta}$. Explicitly, $\widetilde{X}_{\!\scriptscriptstyle\beta_{n}}=i\partial_{n}H_{\!\scriptscriptstyle\beta}$, $\widetilde{X}_{\!\scriptscriptstyle\beta_{\bar{n}}}=-i\partial_{\bar{n}}H_{\!\scriptscriptstyle\beta}$. Here, we have each fixed $\left|\phi_{\scriptscriptstyle 0}\right\rangle$ gives an evaluation map $[\left|\phi_{\scriptscriptstyle 0}\right\rangle]$ that determines the complex number value for each expression in $(H_{\!\scriptscriptstyle\beta},\widetilde{X}_{\!\scriptscriptstyle\beta_{n}},\widetilde{X}_{\!\scriptscriptstyle\beta_{\bar{n}}},\widetilde{K}_{\!\scriptscriptstyle\beta_{m\bar{n}}}\!\!=\!{\nabla}_{\!\!{{m}}}\widetilde{X}_{\!\scriptscriptstyle\beta_{\bar{n}}}$) for all elements $\beta$ of the observable algebra. Note that each expression, e.g. $\widetilde{X}_{\!\scriptscriptstyle\beta_{\bar{n}}}$ is a function of the coordinates which have fixed real number values for $\left|\phi_{\scriptscriptstyle 0}\right\rangle$. The map $[\left|\phi_{\scriptscriptstyle 0}\right\rangle]$ evaluates, or assigns a noncommutative value, to each specific observable $\beta$ as given here as the particular set of infinite number of complex numbers which is really an element of a noncommutative algebra. We are on our way to give the product within the latter algebra, one that gives the noncommutative value of $\beta\gamma$ from those of $\beta$ and $\gamma$, for the same state vector $\left|\phi_{\scriptscriptstyle 0}\right\rangle$. Note that $\beta(\hat{p}^{i},\hat{x}^{i})$ are in general ‘complex functions’, meaning non-Hermitian operators, essentially complex linear combinations of Hermitian operators, making up the observable algebra as a $C^{*}$-algebra. Therefore, $H_{\!\scriptscriptstyle\beta}$ is in general complex too. The real ones correspond to the Hermitian operators. The same holds for the $f_{\!\scriptscriptstyle\beta}$ functions.111 For the preferred formulation within our perspective of quantum relativity [3], we have representation of the coordinate operators $\hat{p}_{i}$ and $\hat{x}_{i}$ as ${p}_{i}\star=p_{i}-i\partial_{\!x^{i}}$ and ${x}^{i}\star=x_{i}+i\partial_{\!p^{i}}$, respectively, on the space of coherent state wavefunctions $\phi(p^{i},x^{i})$, with $\beta(\hat{p}^{i},\hat{x}^{i})$ given by $\beta(p^{i},x^{i})\star$, with the $\star$ as the one in the Moyal star product (here in $\hbar=2$ units). The latter is Hermitian for a real function $\beta(p^{i},x^{i})$ and non-Hermitian for a complex one. Actually, the operator product $\beta(p^{i},x^{i})\!\star~{}\gamma(p^{i},x^{i})\star$ is then exactly $[\beta(p^{i},x^{i})\star\gamma(p^{i},x^{i})]\star$. We have, for the Kähler manifold with $H_{\!\scriptscriptstyle\beta}$ functions generating Hamiltonian flows as isometries, ${\widetilde{K}}_{\!\scriptscriptstyle\beta_{n\bar{m}}}=-{\widetilde{K}}_{\!\scriptscriptstyle\beta_{\bar{m}n}}=-i{\partial}_{{{n}}}{\partial}_{\bar{{m}}}H_{\!\scriptscriptstyle\beta}$ and ${\widetilde{K}}_{\!\scriptscriptstyle\beta_{m{n}}}={\widetilde{K}}_{\!\scriptscriptstyle\beta_{\bar{m}\bar{n}}}=0$. The Kähler product gives

[TABLE]

The set $\{\widetilde{X}_{\!\scriptscriptstyle\beta_{n}},\widetilde{X}_{\!\scriptscriptstyle\beta_{\bar{n}}},\widetilde{K}_{\!\scriptscriptstyle\beta_{m\bar{n}}}\}$ is really the set of all independent nonzero derivatives of $H_{\!\scriptscriptstyle\beta}$; $\widetilde{X}_{\!\scriptscriptstyle\beta_{\bar{n}}}$ being the conjugate of $\widetilde{X}_{\!\scriptscriptstyle\beta_{\bar{n}}}$ only for the Hermitian $\beta$. The second order derivatives give the matrix elements, $\widetilde{K}_{\!\scriptscriptstyle\beta_{m\bar{n}}}=-\frac{i}{2\hbar}\left\langle n|\beta|m\right\rangle$. They are not dependent on the state at all, and the $\widetilde{K}_{\!\scriptscriptstyle\beta\gamma_{\,m\bar{n}}}$ expression above simply gives the matrix elements of $\beta\gamma$ in terms of the matrix elements of $\beta$ and $\gamma$. The latter alone gives an isomorphic description of the observable algebra. The first order derivatives are, however, linear functions of the state coordinates, in particular $\widetilde{X}_{\!\scriptscriptstyle\beta_{n}}=\frac{i}{2\hbar}\sum_{m}\bar{z}^{m}\left\langle m|\beta|n\right\rangle$. Putting together the values of the first and second derivatives, say $\widetilde{X}_{\!\scriptscriptstyle\beta_{n}}$ and $\widetilde{K}_{\!\scriptscriptstyle\beta_{m\bar{n}}}$, may allow us to obtain the values of the coordinates and hence determine the state vector $\left|\phi\right\rangle$ up to an overall phase on which the $H_{\!\scriptscriptstyle\beta}$ functions have no dependence. That is feasible for any invertible operator in itself.

We are more interested in results for the $f_{\!\scriptscriptstyle\beta}$ functions. Unlike the Hilbert space function $H_{\!\scriptscriptstyle\beta}$, $f_{\!\scriptscriptstyle\beta}$ as a function on the projective Hilbert space has (affine) coordinates $w^{n}$. However, expressions in terms of the $z^{n}$ as homogeneous coordinates are still useful. Hence we present both. Note that though a set of affine coordinates can be easily given from a set of homogeneous coordinates, the correspondence is not unique and no set of affine coordinates in itself can cover the whole projective Hilbert space as a geometric manifold. As an $f_{\!\scriptscriptstyle\beta}$ function is really independent of the normalization of $\left|\phi\right\rangle$, using the $z^{n}$ coordinates for a $\left|\phi_{\scriptscriptstyle 0}\right\rangle$ to determined an evaluation map $[\left|\phi_{\scriptscriptstyle 0}\right\rangle]$ has only an overall complex phase ambiguity as a common factor to all $z^{n}$. Otherwise, it corresponds uniquely to a physical state as a point in the projective Hilbert space specified by the $w^{n}$ coordinates. The definite evaluation map $[\phi_{\scriptscriptstyle 0}]$ for such a physical state is free from such ambiguity, and evaluates by applying fixed $w^{n}$ coordinates.

Using the $z^{n}$ coordinates with $\tilde{\omega}^{m\bar{n}}=-i\tilde{g}^{m\bar{n}}$, we have

[TABLE]

Using the $w^{n}$ coordinates (no $n=0$) with ${\omega}^{m\bar{n}}=-i{g}^{m\bar{n}}$, we have

[TABLE]

where the index for the components, as the coordinates, runs from one but the summations includes $m=0$ with $w^{\scriptscriptstyle 0}=\bar{w}^{\scriptscriptstyle 0}=1$. The similar form of the two sets of the results is deceiving. Apart from the different coordinate derivatives, denoted by the same $\partial_{n}$ symbol for simplicity (with a pair less in the second case), the Hamiltonian vector field ${\tilde{X}}_{\!\scriptscriptstyle\beta}$ is obtained from the symplectic form $-i\tilde{g}^{m\bar{n}}$ while ${{X}}_{\!\scriptscriptstyle\beta}$ from the symplectic form $-i{g}^{m\bar{n}}$, though for the same $f_{\!\scriptscriptstyle\beta}$ function. Actually, ${\tilde{X}}_{\!\scriptscriptstyle\beta}$ and $-i\tilde{g}^{m\bar{n}}$ are, as tensors, the horizontal lifts of ${{X}}_{\!\scriptscriptstyle\beta}$ and $-i{g}^{m\bar{n}}$, respectively. Their covectors are identical though, i.e. $\tilde{X}_{\!\scriptscriptstyle\beta_{n}}dz^{n}={X}_{\!\scriptscriptstyle\beta_{n}}dw^{n}$ (sum without $n=0$ for the right hand side), because the ${{X}}_{\!\scriptscriptstyle\beta}$ covector is horizontal [1]. Applying the Kähler product, we have

[TABLE]

and

[TABLE]

The above results illustrate what is said above, that the symmetry data for any two observables have a product which gives the symmetry data of the product observable – almost but not exactly. The elements of the metric and the inverse metric tensor are involved too. Upon a more careful thinking, however, that feature is not unreasonable, in fact it is quite normal. In a non-Euclidean space, or even in an Euclidean space depicted in non-Cartesian coordinates, even the algebraic relations between classical observables commonly involve the metric. For example, the energy of a free Newtonian particle with the momentum components $p_{\!\scriptscriptstyle r}$, $p_{\!\scriptscriptstyle\theta}$, and $p_{\!\scriptscriptstyle\psi}$ in spherical coordinates is $E=\frac{1}{2m}\!\left(p_{\!\scriptscriptstyle r}^{2}+\frac{1}{r^{2}}p_{\!\scriptscriptstyle\theta}^{2}+\frac{1}{r^{2}\sin^{2}\!\theta}p_{\!\scriptscriptstyle\psi}^{2}\right)$. Generally, if the physical space is a curved manifold with metric $g_{ab}$, we have $E=\frac{1}{2m}g_{ab}p^{a}p^{b}$. The metric is in fact always there, only that for the case of an Euclidean geometry in the standard Cartesian coordinates, its values are simply $\delta_{ab}$ at every point and hence we can write the expression without showing it explicitly. Here for calculating the symmetry data for any operator product through the $f_{\!\scriptscriptstyle\beta}$ functions, however, we always need the values of the metric and inverse metric tensor elements at the phase space point. The truth is that the metric tensor elements are also all over Eqs.(5) and (8) implicitly, as in much the same form as they explicitly show up in Eq.(9). The other apparent problem is that the expressions in terms of the $z^{n}$ coordinates also depend on $|z|^{2}$, but it we can easily avoided by looking only at the normalized $\left|\phi\right\rangle$. In our formulation with $z^{n}$ bearing the physical dimension of length, the preferred normalization is actually $|z|^{2}=2\hbar$ at which $f_{\!\scriptscriptstyle\beta}=H_{\!\scriptscriptstyle\beta}$ [1].

The next thing to examine is if the symmetry data for an observable on a fixed state $[\phi_{\scriptscriptstyle 0}]$ can be determined experimentally, at least in principle. We consider first a Hermitian $\beta$ and the $H_{\!\scriptscriptstyle\beta}$ functions first. For a normalized state, $H_{\!\scriptscriptstyle\beta}([\phi_{\scriptscriptstyle 0}])$ (and $f_{\!\scriptscriptstyle\beta}([\phi_{\scriptscriptstyle 0}])$) is just the familiar expectation value of an operator $\beta$. $\widetilde{K}_{\!\scriptscriptstyle\beta_{m\bar{n}}}$ are the matrix elements on a supposedly known set of orthonormal states, hence have no problem either. With those, determining $\widetilde{X}_{\!\scriptscriptstyle\beta_{n}}([\phi_{\scriptscriptstyle 0}])$ means determining $\widetilde{X}_{\!\scriptscriptstyle\beta_{n}}(\phi_{\scriptscriptstyle 0})$ for the normalized $\left|\phi_{\scriptscriptstyle 0}\right\rangle$ up to an overall phase factor as the one among the $z^{n}$ coordinates. The phase is certainly not to be determined. More specifically, if we take as the basis $\left|z_{n}\right\rangle$ the set of eigenstates for $\beta$, assuming no degeneracy, the nonzero $\widetilde{K}_{\!\scriptscriptstyle\beta_{m\bar{n}}}$ are exactly $\widetilde{K}_{\!\scriptscriptstyle\beta_{n\bar{n}}}=\frac{-i\lambda_{n}}{2\hbar}$ with the eigenvalues $\lambda_{n}$ and we simply have $\widetilde{X}_{\!\scriptscriptstyle\beta_{n}}=\frac{i\lambda_{n}}{2\hbar}\bar{z}^{n}$ (no sum). Gone are the days when people debate if the physical quantum state can be determined or observed. On the one hand, we have theoretical analyses, most notably the line of work presented in Ref.[7], illustrating the involved mathematics; on the other there has been developed experimental efforts in quantum optics [8], especially the technique of optical homodyne tomography [9], to actually measure the state. Of course, any practical measurement gives only approximations. Then, the $z^{n}$ coordinates for a normalized $\left|\phi_{\scriptscriptstyle 0}\right\rangle$, up to an overall phase factor, and $\widetilde{X}_{\!\scriptscriptstyle\beta_{n}}$ (and $\widetilde{X}_{\!\scriptscriptstyle\beta_{\bar{n}}}$ as the complex conjugate) can be determined in principle. For non-Hermitian $\beta$, we can take it as a ‘complex function’ with real and imaginary parts as Hermitian operators (see the above footnote). With the linear structure of the algebra well-preserved in the Kähler product algebra and the symmetry data algebra, it is then a trivial extension to incorporate the case of such a non-Hermitian $\beta$. The symmetry data in terms of an $f_{\!\scriptscriptstyle\beta}$ function, given in Eqs.(6) and (7), then clearly pose no further qualitative difficulty. $\tilde{X}_{\!\scriptscriptstyle\beta_{n}}([\phi_{\scriptscriptstyle 0}])$ and $\tilde{X}_{\!\scriptscriptstyle\beta_{\bar{n}}}([\phi_{\scriptscriptstyle 0}])$ or $\tilde{X}_{\!\scriptscriptstyle\beta_{n}}([\left|\phi_{\scriptscriptstyle 0}\right\rangle])$ and $\tilde{X}_{\!\scriptscriptstyle\beta_{\bar{n}}}([\left|\phi_{\scriptscriptstyle 0}\right\rangle])$, up to the overall phase factor in the $z^{n}$ coordinates, as well as $\tilde{K}_{\!\scriptscriptstyle\beta_{m\bar{n}}}([\phi_{\scriptscriptstyle 0}])$, ${X}_{\!\scriptscriptstyle\beta_{n}}([\phi_{\scriptscriptstyle 0}])$, ${X}_{\!\scriptscriptstyle\beta_{\bar{n}}}([\phi_{\scriptscriptstyle 0}])$, and ${K}_{\!\scriptscriptstyle\beta_{m\bar{n}}}([\phi_{\scriptscriptstyle 0}])$ can all be determined likewise. The kind of expression like ${X}_{\!\scriptscriptstyle\beta_{n}}([\phi_{\scriptscriptstyle 0}])$ means only the complex number of ${X}_{\!\scriptscriptstyle\beta_{n}}$ after the fixed values of the coordinates, $w^{n}$ or $z^{n}$ are fixed for the ray of the state vector $\left|\phi_{\scriptscriptstyle 0}\right\rangle$. Note that the values of $\tilde{K}_{\!\scriptscriptstyle\beta_{m\bar{n}}}$, ${X}_{\!\scriptscriptstyle\beta_{n}}$, ${X}_{\!\scriptscriptstyle\beta_{\bar{n}}}$, and ${K}_{\!\scriptscriptstyle\beta_{m\bar{n}}}$ are in general independent of the undetermined phase factor, also in $w^{n}$ coordinates. Moreover, knowing the coordinates, we have all the elements of the metric tensors at the state, which can be considered as a theoretical input at this point. From the perspective of Refs.[2, 3], the metric gives a real number distance between the two quantum particle states, the two points in the quantum model of the physical space, and its connection to quantum observables is explored in Ref.[1]. One would like to find a way to measure or verify the metric tensor experimentally but we will not go more into that here. It is important to note the following. While an $f_{\!\scriptscriptstyle\beta}$ function, unlike the $H_{\!\scriptscriptstyle\beta}$ function, has nontrivial higher derivatives, they have however definite relation to the first and second order ones through the metric [5] as a result of the Kählerian nature of the $f_{\!\scriptscriptstyle\beta}$ function. That goes for both the descriptions in terms of the $z$- or $w$-coordinates. Hence ${X}_{\!\scriptscriptstyle\beta_{n}}$ , ${X}_{\!\scriptscriptstyle\beta_{\bar{n}}}$, and ${K}_{\!\scriptscriptstyle\beta_{m\bar{n}}}$, or $\tilde{X}_{\!\scriptscriptstyle\beta_{n}}$ , $\tilde{X}_{\!\scriptscriptstyle\beta_{\bar{n}}}$, $\tilde{K}_{\!\scriptscriptstyle\beta_{m\bar{n}}}$, make up the complete set of independent derivatives, or covariant derivatives. The fact that the operator, or its representation $f_{\!\scriptscriptstyle\beta}$ or $H_{\!\scriptscriptstyle\beta}$, can indeed be determined, at least locally, by the values of all its derivatives at a point is simply the mathematics of the Taylor series expansion.

3 Physics Discussions

We have argued above that the symmetry data for an operator at a fixed physical state can in principle be experimentally determined. Note that it is actually most directly represented by the set of complex numbers $\{f_{\!\scriptscriptstyle\beta}([\phi_{\scriptscriptstyle 0}]),{X}_{\!\scriptscriptstyle\beta_{n}}([\phi_{\scriptscriptstyle 0}]),{X}_{\!\scriptscriptstyle\beta_{\bar{n}}}([\phi_{\scriptscriptstyle 0}]),{K}_{\!\scriptscriptstyle\beta_{m\bar{n}}}([\phi_{\scriptscriptstyle 0}])\}$. The corresponding set in terms of the homogeneous coordinates $z^{n}$, and even the representation through the $H_{\!\scriptscriptstyle\beta}$ function and its derivatives, have the complication of having to be restricted to the normalized state vectors, with an unphysical overall phase factor for the $z^{n}$ coordinates formally involved in the first derivatives. That does no harm. In fact, we can even take a conventional definition of the coordinates to fix it, for example taking $z^{\!\scriptscriptstyle 0}$ as real. Unlike the $w^{n}$ coordinates, the $z^{n}$ coordinates cover the whole space of physical states, $\mathcal{P}$. All that, however, is only of a theoretical interest. For the idea of the noncommutative (number) value of an observable to be of any practical use, one does not want to determine them as indirectly as discussed above. We want their direct measurements. We argue below that such an idea, although going much beyond the familiar, may not be as crazy as it may sound to those who never thought about it before.

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 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 the 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. We are interested in finding the better way to model those ‘values’. To consider the full practical implementations of what is here theoretically discussed, one may have to devise an appropriate apparatus and calibrate its output with the noncommutative values.

Taking a step backward to the conservative consideration of the von Neumann measurements on a quantum observable, one must note that a single real eigenvalue outcome in itself hardly gives any information about the “value” of the observable on a particular state. The best real number representation of the value is the expectation value as the mean of a sufficient statistical collection of the eigenvalue outcomes from repeated measurements. But then it has a standard statistical uncertainty. However, we have more than the expectation value and the uncertainty. We have the whole distribution which contains its full information in terms of an infinite number of real numbers in all the moments. Depending on the desired precision, we can take enough statistics and calculate a large enough number of the lower moments to get to an approximation of the full information content for the value of the observable on the fixed state. There is no reason, other than ignorance, to discard all those real number information in all the moments and use only one or two as done in a standard discussion about quantum observables. The full infinite set of real numbers, or the whole distribution, should be taken as the value of the observable on a state. The latter has an obvious parallel with the notion of the symmetry data. Again, the value of a quantum observable for a state should be represented by the element of a noncommutative algebra rather than by the real numbers, i.e. of a commutative one. Such an element can also be represented by an infinite set of the real/complex numbers.

The representation of our noncommutative value for an observable in terms of the symmetry data as the infinite set of complex numbers has the first number being the functional value of $f_{\scriptscriptstyle\beta}$, exactly the expectation value. It is real for a Hermitian $\beta$, and is no doubt the best single real number representation of the value of $\beta$ as a physical quantity. What the full noncommutative value represents is the full information the theory can give mathematically matching any observable to a fixed physical state. It goes beyond determining the whole statistical distribution of results from von Neumann measurements. We have seen above that the set of complex numbers essentially contains enough information to determined the exact physical state as well as all matrix elements of the observable with respect to the fixed coordinate basis. And the noncommutative value for $\beta$ and $\gamma$ allows us to determine that for the products $\beta\gamma$ and $\gamma\beta$, satisfying whatever commutation relation they have.

Intuitively, when the state is fixed, all physical properties should be fixed. To have the idea implemented in quantum mechanics is workable, we simply have to go beyond the idea of describing each simple physical property, such as a position coordinate, by a single real number. In our opinion, the stubborn physicists’ attachment to the real number values is what makes quantum theory seemingly counter-intuitive. We can and should go beyond that. There is no hidden variables in quantum mechanics and the noncommutative variables of position and momentum fully describe the states, but the full description of their values for a definite state has been ‘hidden’ beyond any single real number representation for each variable. The latter gives an incomplete description, while the complete information is always there in the noncommutative values. The mathematician Takesaki talked about operator algebras as ‘a number theory in analysis’ [10]. From a physics point of view, observables are represented by the operators, the variables to be evaluated on a physical state. The notion of their noncommutative values may be really be seen as noncommutative numbers. A proper representation of a $C^{*}$-algebra, which is what physicists should focus on as a candidate for the observable algebra [11], gives an operator algebra on a Hilbert space. The corresponding projective Hilbert space is a space of pure states which is a mathematical object dual to (the representation of) the $C^{*}$-algebra [12, 13]. $C^{*}$-algebras are noncommutative geometric objects [14]. A $C^{*}$-algebra of the quantum operators, as functions of the six position and momentum operators [3], may be taken as having them as coordinate observables of the geometry whose noncommutative values for each physical state may plausibly be seen as an alternative description of an infinite number of real/complex number coordinates of the projective Hilbert space. The study of that idea is reported in a companion paper Ref.[1].

**Acknowledgments  ** We thank Suzana Bedić for helping to improve the language of the presentation. The authors are partially supported by research grants number 109-2112-M-008-016 and 107-2119-M-008-011 of the MOST of Taiwan.