1 Introduction↩︎

What defines a metric, that is, a way to measure distances on a given manifold? In the case of classical, i.e. non-quantum physics, the answer is provided by the general of relativity (GR). Metrics are determined by the distribution of masses and currents of matter. The modern approach to this problem began with Clifford’s work [1] in 1876. Thirty-nine years later, Einstein gave the solution in the form of the general theory of relativity.

The concept of a metric is basic one and enters almost every equation of physics. For example, in order to define one of the basic in physics operators, the Laplace operator, it is necessary to determine a metric first.

In the case of quantum physics whose states are defined in Hilbert space, there is no such a single measure of distance. On the contrary, there are many measures of distance between states, some used in quantum information theory (also applied in research related to quantum gravity, see e.g. [2]) but none of them follows from some fundamental principle. Since Hilbert space is a vector complex space, so one can introduce a metric that is induced from \(C^{N}\)and call this metric canonical. Giving as a result the Fubini-Study (FS) metric. And based on it to determine the distance. In the case when Hilbert space is represented by square-integrable functions \(L^{2}\left( M\right)\) on some manifold \(M\), then the scalar product on this Hilbert space is given by the volume form \(d\mu\)on \(M\). Again, this scalar product leads to the FS metric. For both finite \(N\)and for \(L^{2}\left( M\right)\), the largest distance between points of these Hilbert spaces is normalized to \(\pi\). However, such a canonical metric does not represent the complexity of the quantum system. Nevertheless, and as is well known, the probabilistic interpretation of the quantum system is based on this metric.

Another issue is the geometrization of thermodynamics. There are such metrics as Weinhold metric or Ruppeiner metric, where the relationship between them is found. However, also in this case none of the metrics is derived from some fundamental equation. Since widely studied and the well-known topic is the thermodynamics of black holes, the combination of geometrization of thermodynamics and black hole seems promising. Many papers have been written on this topic, e.g. [3-5].

The fundamental classical concept describing a physical system is the action integral. The energy-momentum tensor of a system results as a variation of the action integral with respect to a metric. The general relativity links the energy-momentum tensor to the geometric properties of space-time, i.e., the Riemann tensor. In contrast, the quantum concept describing the system is the density matrix. And there is no theory or relationship that links the density matrix to the quantum properties of space-time, because such concepts do not exist. One may ask: are there any hints in existing theories that would provide such ideas?

The first such a hint may be the concept of entropy determined by the density matrix. One should look for an extension of the concept of entropy in an analogous way to how energy density and momentum density combine in the energy-momentum tensor.

A second such hint is geometrodynamics [6]. One way to find a quantum theory of gravity is to try to quantize the general theory of relativity using canonical quantization. In this approach, the Wheeler-De Witt (WDW) equation plays a fundamental role. Many interesting results have been obtained following this way of reasoning, e.g.[7-9]. So, the concepts that allow to describe quantum space-time should be obtained in the framework of geometrodynamics from the WDW equation.Although this is not a work on quantum gravity (QG), and is only an attempt to find an analogy of the equivalence principle in quantum mechanics, we will just mention the two most common and fruitful approaches to the issue of QM as a metric theory. That means canonical and covariant approachs. In the 1960s, De Witt presented attempts to quantize the gravitational field in canonical and covariant approaches [10-12]. Since then, much has been understood and achieved but many problems remain.The actual question is whether these approaches are equivalent or not, namely they possibly represent a single formal theory, or not. Otherwise, the issue should be that of finding out which of them, if any - simply on the basis of general physical principles - may appear as the correct one. Regarding possible alternative routes to QG theory, covariant quantization should provide hints to the considered problem. The literature on covariant quantization of gravity is vast, so the authors will cite only one [13] and another that is a new approach to the Hamiltonian formulation of gravity [14-15].

Moreover, in quantum field theory in curved space-time (the zero approximation of quantum gravity), the problem of determining the vacuum state of the field arises. There are many vacuum states which are not unitary related. For example in the Schwarzschildspacetime with free scalar field there are three well-known vacuum states: Hartle-Hawking, Boulware and Unruh. Each of these gives different expectation values of the field operators [16]. Another example is de Sitter spacetime with families of vacuua states of quantum scalar field [17]. So the natural question is about the relationship between such states. A good measure to determine these relationships is the distance between them.

In this paper we propose a metric in a general case of a Hilbert space of a quantum mechanical system. This metric is infinitely dimensional version of Fisher-Rao metric on infinitly dimensional sphere \(S^{\infty }\). We apply such a metric for simplest quantum systems: a free particle and harmonic oscillator. The "distance" given by the relative entropy is derived and calculated for different quantum systems.

The paper is organized as follows. In the section 2 it is recalled how the distance is determined in classical physics. In the section 3, the metrics in the space of probability distributions are presented and the Fisher-Rao (FR) metric is derived as a condition for the stationarity of the "action integral". In the section 4 we give the metric in the Hilbert space of a quantum mechanical system. In the section 5 we find the FR metrics in the case of a free particle and harmonic oscillator. In the section 6 we present and calculate the "distance" given by the relative entropy. The section 7 is devoted to conclusions.

2 Measure of the distance in space-time↩︎

To determine the distance in space-time \(M\) with fixed symmetry and matter with a given energy-momentum tensor \(T_{\mu \nu }\), it is necessary to solve Einstein’s equations:\[G_{\mu \nu }\left( g\right) =16\pi T_{\mu \nu }\left( g\right) \]that is, to determine the metric field \(g\) where \(G_{\mu \nu }\) is the Einstein tensor. This is a nontrivial task and there are few exact solutions. From the metric "\(g\)" obtained in this way, it is necessary to find the line (1-dimensional submanifold) \(\gamma\) along which the distance will be measured. The line \(\gamma\) is parameterized by the affine parameter "\(s\)":\[\gamma =\left\{ x\in M:\text{ }x=x\left( s\right) \right\} \]and "\(x\)" are coordinates on \(M\). The equation of such a line, is found from the stationarity of the functional:\[\delta L=0,\]where:\[L\left[ \gamma \right] =\frac{1}{2}\dint\limits_{s_{1}}^{s_{2}}g_{\mu \nu }\left( x\right) \frac{dx^{\nu }}{ds}\frac{dx^{\nu }}{ds}ds \]with the fixed boundary points: \(x_{1}=x\left( s_{1}\right)\) and \(x_{2}=x\left( s_{2}\right)\). It leads to the geodesic equation:\[\frac{d^{2}x^{\mu }}{ds^{2}}+\Gamma _{\nu \rho }^{\mu }\frac{dx^{\nu }}{ds}\frac{dx^{\rho }}{ds}=0, \]with the boundary conditions \(x_{1}=x\left( s_{1}\right)\) and \(x_{2}=x\left( s_{2}\right)\). In order to solve this equation in an unambiguous way, it is necessary to give initial conditions:\[x\left( s_{1}\right) =x_{1}\text{ \;and }\left. \frac{dx}{ds}\right\vert _{s=s_{1}}=\overset{\cdot }{x}_{1}. \]Thus one has to express these initial conditions by the boundary conditions. The expression of boundary conditions by initial conditions is unambiguous when the points \(x_{1}\) and \(x_{2}\) are "close". The expression "close" means that \(x_{1}\) and \(x_{2}\) are not conjugate. Finally, the distance \(d\) between points \(x_{1}\) and \(x_{2}\) is given by the integral:\[0<d\left( x_{1},x_{2}\right) =\dint\limits_{s_{1}}^{s_{2}}\sqrt{\left\vert g_{\mu \nu }\frac{dx^{\nu }}{ds}\frac{dx^{\nu }}{ds}\right\vert }ds.\] Since the metric \(g\) at each point has a signature \((-,+,+,+)\), there are spatial-like \(d^{2}\left( x_{1},x_{2}\right) >0\), time-like \(d^{2}\left( x_{1},x_{2}\right) <0\) and light-like \(d^{2}\left( x_{1},x_{2}\right) =0\) distances. So the geodesic is a line of maximum length. A line with a minimum length of zero between two points (causally related) always exists, and it is the sum of zero geodesics.

3 Metrics for space of probability distributions↩︎

For a random variable \(X\in \Omega\) (where the set \(\Omega\) can be continuous ordiscrete) is given a probability distribution \(p\):\[p=p\left( X;\theta _{1},...,\theta _{N}\right) , \]where the numbers \(\left( \theta _{1},...,\theta _{N}\right) \equiv \Theta\) are the parameters of the distribution \(p\) belonging in the general case to some manifold \(M\). So \(p\) is the function defined on the Cartesian product \(\Omega \times M\) with values between 0 and 1 for each \(\Theta\): \[p:\Omega \times M\rightarrow \lbrack 0,1]. \]Moreover there is the normalization condition:\[\int_{\Omega }p\left( X;\Theta \right) d\mu =1, \]where \(d\mu\) is measure on \(\Omega\). This condition defines the manifold \(M\). So the given probability distribution \(p\) determines \(M\) but, on the other hand, this distribution itself depends on the points of \(M\). Thus, the space of probability distributions is obtained:\[\mathcal{P}=\left\{ p\left( X;\Theta \right) \right\} \]as the space of functions over \(M\). The question arises: how can the distance between two distributions on \(\mathcal{P}\) be determined? And the question: is there any distinguished set of metrics on \(M\)?

In the geometric approach, the metric is given by the relation (e.g ):\[g_{ab}\left( \Theta ;F\right) =\int_{\Omega }p\frac{\partial F}{\partial \theta ^{a}}\frac{\partial F}{\partial \theta ^{b}}d\mu =\int_{\Omega }p\left[ F^{\prime }\left( p\right) \right] ^{2}\frac{\partial p}{\partial \theta ^{a}}\frac{\partial p}{\partial \theta ^{b}}d\mu , \]where \(F\) is a function of \(p\) and prime means differentiation with respect to \(p\). From this equation one can obtain an "action integral" \(S\) for the function \(F\) depending on the one degree of freedom \(p:\)\[S[F]=\dint L\left[ F;p\right] dp,\]where the Lagrangian \(L\left[ F;p\right]\) for this action is equal to:\[L\left[ F;p\right] =p\left[ F^{\prime }\left( p\right) \right] ^{2}.\] Hence, the stationarity condition for this "action integral" leads to the Euler-Lagrange equation:\[\frac{d}{dp}\left( pF^{\prime }\right) =0, \]with the solution:\[F\left( p\right) =k\ln p+F_{0}, \]where \(k\) and \(F_{0}\) are constants of integration. In this way, the metric for this solution is of the form:\[g_{ab}\left( \Theta \right) =k^{2}\int_{\Omega }p\left( X;\Theta \right) \frac{\partial \ln p}{\partial \theta ^{a}}\frac{\partial \ln p}{\partial \theta ^{b}}d\mu =k^{2}\int_{\Omega }\frac{\partial _{a}p\partial _{b}p}{p}d\mu . \]This is the Fisher-Rao (FS) metric. The other form of this metric is as follows:\[g_{ab}\left( \Theta \right) =-k^{2}\int_{\Omega }p\partial _{ab}^{2}\ln pd\mu . \]Hence, the infinitesimal square of length on \(M\) has the form:\[dl^{2}=g_{ab}d\theta ^{a}d\theta ^{b} \]

As an example, we will consider the Gauss distribution: \[p\left( X;\theta ^{1},\theta ^{2}\right) =\frac{1}{\sqrt{2\pi }\theta ^{1}}\exp \left[ -\frac{\left( X-\theta ^{2}\right) ^{2}}{2\left( \theta ^{1}\right) ^{2}}\right] . \]Thus the FR metric is well-known and equal to (for \(k=1\)):\[dl^{2}=\frac{1}{\left( \theta ^{1}\right) ^{2}}\left( d\left( \theta ^{2}\right) ^{2}+2d\left( \theta ^{1}\right) ^{2}\right) . \]This is the metric on the Poincare upper half plane \(\mathbf{H}\) defined by the condition \(\theta ^{1}>0\). Thus the geodesic distance between two Gauss distributions:\[p_{1}=p\left( X;\Theta _{1}\right) \text{ and }p_{2}=p\left( X;\Theta _{2}\right) \]in this metric is equal to:\[d\left( p_{1},p_{2}\right) =2\sinh ^{-1}\left[ \frac{\left\vert \Theta _{2}-\Theta _{1}\right\vert }{2\sqrt{\theta _{2}^{1}\theta _{1}^{1}}}\right] , \]where \(\Theta _{a}=\left( \theta _{a}^{1},\theta _{a}^{2}\right)\), \(a=1,2\), \(\left\vert \Theta \right\vert =\sqrt{\left( \theta ^{1}\right) ^{2}+\left( \theta ^{2}\right) ^{2}}\) and \(\sinh ^{-1}X=\ln \left( X+\sqrt{1+X^{2}}\right)\). If we omit the stationarity condition, the metric for the Gauss distribution takes the form:\[g_{11}=\frac{\sqrt{2}}{\theta ^{1}}\int_{\mathbf{R}^{1}}p^{3}\left( z\right) \left[ F^{\prime }\left( p\right) \right] ^{2}\left[ 1-4z^{2}+4z^{4}\right] dz, \]\[g_{22}=\frac{1}{\sqrt{2}\theta ^{1}}\int_{\mathbf{R}^{1}}p^{3}\left( z\right) \left[ F^{\prime }\left( p\right) \right] ^{2}z^{2}dz, \]\[g_{12}=0, \]where:\[p\left( z\right) =\frac{1}{\sqrt{2\pi }\theta ^{1}}\exp \left( -z^{2}\right) . \]If we require that the following condition is satisfied:\[p^{3}\left[ F^{\prime }\left( p\right) \right] ^{2}=p, \]then \(F\left( p\right) =\ln p\) and again the FR metric is obtained. So one can say that for the Gaussian distribution the stationarity condition and above condition (which can be referred to as a simplicity condition) give the same result, namely FR metric.

4 FR metric in quantum mechanics↩︎

Quantum mechanics provides probability distributions \(P\) expressed by wave functions \(\psi\). In general, the wave function \(\Psi\) depends on time \(t\), the parameters of the system \(\omega\) and the initial state \(\psi\), \[\Psi =\Psi \left( x,t,\omega ,\psi \right) \]where \(x\) denote the spatial coordinates. If \(\widehat{H}\) is time independent Hamiltonian of the system, then in the eigenbasis \(\psi _{n}\) of \(\widehat{H}\) with eigenvalues \(E_{n}\) the initial state \(\psi =\dsum\limits_{n=0}c_{n}\psi _{n}\left( x;\omega \right)\) evolves as follows:\[\Psi \left( x,t,\omega ,c\right) =\exp \left( -it\widehat{H}\right) \psi =\dsum\limits_{n}c_{n}\exp \left( -itE_{n}\left( \omega \right) \right) \psi _{n}\left( x;\omega \right) . \]The complex coefficients \(c=\left( c_{n}\right)\) are normalized: \(\dsum\limits_{n=0}\left\vert c_{n}\right\vert ^{2}=1\). The wave function \(\Psi\) can also be expressed via: \[\Psi \left( x;t,\omega \right) =\int_{\mathbf{R}^{N}}d^{N}yK\left( x,t;y,0\right) \psi \left( y;\omega \right) , \]propagator \(K\left( y,0;x,t\right)\) with the initial condition:\[K\left( x,0;y,0\right) =\delta ^{\left( N\right) }\left( x-y\right)\] and \(N\) is the number degrees of freedom. In basis \(\psi _{n}\) the propagator takes the form:\[K\left( x,t;y,0\right) =\dsum\limits_{n}\exp \left( -itE_{n}\left( \omega \right) \right) \psi _{n}^{\ast }\left( x;\omega \right) \psi _{n}\left( y;\omega \right) \]Thus the probability distribution \(P\) is equal to:\[P\left( x;t,\omega ,c\right) =\dsum\limits_{m,n}c_{m}^{\ast }c_{n}I_{mn}\left( x,t,\omega \right) , \]where:\[I_{mn}\left( x,t,\omega \right) =\int_{\mathbf{R}^{2N}}d^{N}yd^{N}zK^{\ast }\left( x,t;y,0\right) K\left( x,t;z,0\right) \psi _{m}^{\ast }\left( y;\omega \right) \psi _{n}\left( z;\omega \right) =I_{nm}^{\ast }\left( x,t,\omega \right) \]The probability normalization condition is satisfied:\[\int_{\mathbf{R}^{N}}d^{N}xP\left( x;t,\omega ,c\right) =1. \]Thus, for a fixed time \(t\) and parameters \(\omega\), the FR metric is a function of the complex numbers \(c_{m}\) and the metric has the components::\[g_{mn}\left( c\right) =-\int_{\mathbf{R}^{N}}d^{N}xP\frac{\partial ^{2}\ln P}{\partial c_{m}\partial c_{n}}, \]\[g_{\overline{m}n}\left( c\right) =-\int_{\mathbf{R}^{N}}d^{N}xP\frac{\partial ^{2}\ln P}{\partial c_{m}^{\ast }\partial c_{n}}, \]\[g_{\overline{m}\overline{n}}\left( c\right) =-\int_{\mathbf{R}^{N}}d^{N}xP\frac{\partial ^{2}\ln P}{\partial c_{m}^{\ast }\partial c_{n}^{\ast }}=g_{mn}^{\ast }\left( c\right) . \]Thus we get:\[g_{mn}\left( c;t,\omega \right) =\dsum\limits_{k,p}c_{k}^{\ast }c_{p}^{\ast }A_{mn}^{kp}\left( c;t,\omega \right) , \]\[g_{\overline{m}n}\left( c;t,\omega \right) =-\int_{\mathbf{R}^{N}}d^{N}xI_{mn}\left( x,t,\omega \right) +\dsum\limits_{k,p}c_{k}c_{p}^{\ast }A_{kn}^{mp}\left( c;t,\omega \right) ,\] where:\[A_{mn}^{kp}\left( c;t,\omega \right) =\int_{\mathbf{R}^{N}}d^{N}x\frac{I_{km}\left( x,t,\omega \right) I_{pn}\left( x,t,\omega \right) }{P\left( x;t,\omega ,c\right) } \]Since the eigenfunctions \(\psi _{n}\) form a complete and orthogonal system then the first integral on the right-hand side is equal to: \[\int_{\mathbf{R}^{N}}d^{N}xI_{mn}\left( x,t,\omega \right) =\delta _{mn}.\] Thus:\[g_{\overline{m}n}\left( c;t,\omega \right) =-\delta _{mn}+\dsum\limits_{k,p}c_{k}c_{p}^{\ast }A_{kn}^{mp}\left( c;t,\omega \right) . \]The complex numbers \("c"\) form the infinitely-dimensional unit sphere \(S^{\infty }\). Thus the obtained metric is a metric on \(S^{\infty }\) and has the form: \[ds^{2}=g_{mn}dc_{m}dc_{n}+g_{mn}^{\ast }dc_{m}^{\ast }dc_{n}^{\ast }+g_{\overline{m}n}dc_{m}^{\ast }dc_{n}. \]It is real since: \(ds^{2}=\left( ds^{2}\right) ^{\ast }\). The distance between two states given by two sequences \(c=\left( c_{n}\right)\) and \(c^{\prime }=\left( c_{n}^{\prime }\right)\) (which are points on \(S^{\infty }\)) is given by the length of the geodesic \(\gamma\) originating at \(c\) and ending at \(c^{\prime }\). This geodesic is determined by the above metric. The obtained metric (4.10a-b) is the metric on the infinitely dimensional sphere \(S^{\infty }\).

In the next section we will use the above formulas for two quantum systems.

5 Metric for free particle and harmonic oscillator↩︎

As the first example we consider a free quantum particle. The propagator for the free particle (in 1D) of mass \(m\) has the form:\[K\left( x,t;y,0\right) =\sqrt{\frac{m}{2\pi i\hbar t}}\exp \left[ \frac{im\left( x-y\right) ^{2}}{2\hbar t}\right] \]and the wave functions are indexed by the wave vector \(k\):\[\psi _{k}\left( x\right) =\frac{1}{\sqrt{2\pi }}\exp \left( ikx\right) .\] The integrals (4.7) are labeled by the wave vectors \(k\) and \(l\). The system has only one parameter \(m\). Thus inserting above formulas into (4.7) one obtains:\[I_{kl}\left( x;t,m\right) =\frac{m}{\left( 2\pi \right) ^{2}\hbar t}\int_{\mathbf{R}^{2}}dydz\exp \left[ \frac{-im\left( x-y\right) ^{2}}{2\hbar t}+\frac{im\left( x-z\right) ^{2}}{2\hbar t}\right] \exp \left( ikz-ily\right) .\] This integral is easy to compute and is equal to:\[I_{kl}\left( x;t,m\right) =\frac{1}{2\pi }\exp \left[ ix\left( k-l\right) -\frac{i\hbar t}{2m}\left( k^{2}-l^{2}\right) \right] . \]In this way the probability distribution (4.6) is:\[P\left( x;t,m,c\right) =\frac{1}{2\pi }\dsum\limits_{k,l}c_{k}^{\ast }c_{l}\exp \left[ ix\left( k-l\right) -\frac{i\hbar t}{2m}\left( k^{2}-l^{2}\right) \right] , \]where the parameters space \(M\) given by condition (3.3) is infinitly dimensional sphere parametrized by the infinite sequence \(\left( c_{n}\right)\). Thus the metric (4.10a-b) is determined by the integrals (4.11). In the considered case they are given as follow:\[\begin{gather} A_{ln}^{kp}\left( c;t,m\right) =\frac{1}{2\pi }\exp \left[ -\frac{i\hbar t}{2m}\left( k^{2}-l^{2}+p^{2}-n^{2}\right) \right] \times \notag \\ \times \int_{\mathbf{R}^{1}}dx\frac{\exp \left[ ix\left( k-l+p-n\right) \right] }{\dsum\limits_{r,s}c_{r}^{\ast }c_{s}\exp \left[ ix\left( r-s\right) -\frac{i\hbar t}{2m}\left( r^{2}-s^{2}\right) \right] }. \end{gather}\]

As the next example we consider a one-dimensional quantum harmonic oscillator with the energy operator:\[\widehat{H}\left( m,\omega \right) =\frac{1}{2m}\widehat{p}^{2}+\frac{1}{2}m\omega ^{2}\widehat{x}^{2}, \]where \(m\) and \(\omega\) are the mass and frequency respectively. The eigenstates \(\psi _{n}\) and eigenvalues \(E_{n}\) are equal to:\[\psi _{n}\left( x;m,\omega \right) =\frac{1}{\sqrt{2^{n}n!}}\left( \frac{\lambda ^{2}}{\pi }\right) ^{1/4}\exp \left( -\frac{\lambda ^{2}x^{2}}{2}\right) H_{n}\left( \lambda x\right) , \]\[E_{n}=\hbar \omega \left( n+1/2\right) \]where \(\lambda ^{2}=m\omega /\hbar\) and \(H_{n}\) are Hermite polynomials with \(n=0,1,2...\).In this case the wave function \(\Psi \left( x;t,m,\omega \right)\) is obtained from some initial state \(\Psi \left( x;0,m,\omega \right) =\psi \left( x;m,\omega \right) :\)\[\Psi \left( x;t,m,\omega \right) =\int_{\mathbf{R}^{1}}dyK\left( y,0;x,t\right) \psi \left( y;m,\omega \right) , \]where the propagator \(K\) is equal to:\[K\left( x,t;y,0\right) =\frac{\lambda }{\sqrt{2\pi i\sin \left( \omega t\right) }}\exp \left[ \frac{i\lambda ^{2}}{2}\left( x^{2}+y^{2}\right) \cot \left( \omega t\right) -\frac{i\lambda ^{2}xy}{\sin \left( \omega t\right) }\right] \]and:\[K\left( x,0;y,0\right) =\delta \left( x-y\right) .\]Hence the probability distribution \(P\) takes the form:\[P\left( x;t,m,\omega \right) =\int_{\mathbf{R}^{2}}dydzK^{\ast }\left( x,t;y,0\right) K\left( x,t;z,0\right) \psi ^{\ast }\left( y;m,\omega \right) \psi \left( z;m,\omega \right) . \]Finally we get:\[P\left( x;t,m,\omega ,c_{n}\right) =\frac{\lambda ^{2}}{2\pi \sin \left( \omega t\right) }\dsum\limits_{n=0}\left\vert c_{n}\right\vert ^{2}\left\vert I_{n}\left( x;t,m,\omega \right) \right\vert ^{2}, \]where:\[\begin{gather} I_{n}\left( x;t,m,\omega \right) =\frac{1}{\sqrt{2^{n}n!}}\left( \frac{\lambda ^{2}}{\pi }\right) ^{1/4}\exp \left[ -\frac{i\lambda ^{2}x^{2}e^{-i\omega t}}{2\sin \left( \omega t\right) }\right] \times \notag \\ \times \int_{\mathbf{R}^{1}}dyH_{n}\left( \lambda y\right) \exp \left[ \frac{i\lambda ^{2}e^{i\omega t}}{2\sin \left( \omega t\right) }\left( y-xe^{-i\omega t}\right) ^{2}\right] . \end{gather}\]So:\[g_{mn}=-2\delta _{mn}\int_{\mathbf{R}^{1}}dx\left\vert I_{n}\left( x;t,m,\omega \right) \right\vert ^{2}+4c_{m}c_{n}\int_{\mathbf{R}^{1}}dx\frac{\left\vert I_{m}\left( x;t,m,\omega \right) \right\vert ^{2}\left\vert I_{n}\left( x;t,m,\omega \right) \right\vert ^{2}}{P\left( x;t,m,\omega ,c_{n}\right) }. \]The first integral on the right side is equal to:\[\int_{\mathbf{R}^{1}}dx\left\vert I_{n}\left( x;t,m,\omega \right) \right\vert ^{2}=\frac{2\pi }{\lambda ^{2}}\sin \left( \omega t\right) .\] Thus the metric is:\[g_{mn}=-\delta _{mn}\frac{4\pi }{\lambda ^{2}}\sin \left( \omega t\right) +4c_{m}c_{n}\int_{\mathbf{R}^{1}}dx\frac{\left\vert I_{m}\left( x;t,m,\omega \right) \right\vert ^{2}\left\vert I_{n}\left( x;t,m,\omega \right) \right\vert ^{2}}{P\left( x;t,m,\omega ,c_{n}\right) }. \]

As one can see from the above examples, even in the simplest quantum systems the determination of the Fisher-Rao metric on the infinitely dimensional sphere is a nontrivial task. The coefficients of the metric are given by the integrals (4.11).

However if one fixes \(\left( c_{n}\right)\) on \(S^{\infty }\) in the case of oscillator (this procedure can also be applied to the free particle), the parameter space becomes: \[M_{\left( n\right) }=\left\{ \left( m,\omega \right) :m>0\text{ and }\omega >0\right\} \subset \mathbf{R}^{2}. \]The manifold \(M_{\left( n\right) }\) is two dimensional with coordinates given by two positive numbers \(m\) and \(\omega .\) This space corresponds to a set of harmonic oscillators with different masses \(m\) and frequencies \(\omega\) being in the same state given by the sequence \(\left( c_{n}\right)\). Hence the probability distribution related to the eigenstate \(\psi _{n}\) is\[p_{n}\left( x;m,\omega \right) =\frac{1}{2^{n}n!}\frac{\lambda }{\sqrt{\pi }}\exp \left( -\frac{\lambda ^{2}x^{2}}{2}\right) H_{n}^{2}\left( \lambda x\right) \]and the Fisher-Rao metric \(g^{\left( n\right) }\) on \(M_{\left( n\right) }\) obtained from (3.10) has the components (with \(k=1\)):\[g_{mm}^{\left( n\right) }=\frac{1}{2m^{2}}\left( 1-n^{2}-n\right) ,\] \[g_{\omega \omega }^{\left( n\right) }=\frac{1}{2\omega ^{2}}\left( 1-n^{2}-n\right) , \]\[g_{m\omega }^{\left( n\right) }=\frac{1}{4m\omega }\left( 1-n^{2}+3n\right) .\] The line element \(dl_{\left( n\right) }^{2}\) is equal to:\[dl_{\left( n\right) }^{2}=a\frac{dm^{2}}{m^{2}}+a\frac{d\omega ^{2}}{\omega ^{2}}+b\frac{dmd\omega }{m\omega }, \]where \(a=\left( 1-n^{2}-n\right) /2\), \(b=(1-n^{2}+3n)/2\). In the coordinates \(U\) and \(V\) defined as follows: \[U=\ln \left[ \frac{\omega }{\omega _{0}}\left( \frac{m}{m_{0}}\right) ^{b/\left( 2a\right) }\right] , \]\[V=\ln \frac{m}{m_{0}}, \]the metric (5.21) takes the diagonal form: \[dl_{\left( n\right) }^{2}=adU^{2}+\eta \left( n\right) dV^{2} \]with\[\eta \left( n\right) =\frac{1}{8}\frac{\left( 1-n^{2}-5n\right) \left( 3-3n^{2}+n\right) }{1-n^{2}-n}. \]The constants \(m_{0}\) and \(\omega _{0}\) have the mass and \(s^{-1}\) dimensions respectively and have to be determined. Since \(m_{0}\omega _{0}/\hbar\) has dimension \(\left( length\right) ^{-2}\) we make following ansatz:\[\lambda _{0}^{2}=\frac{m_{0}\omega _{0}}{\hbar }=l_{Pl}^{-2}=\frac{c^{3}}{\hbar G},\]where \(l_{Pl}\) is Planck length, so:\[\omega _{0}m_{0}=\frac{c^{3}}{G}.\]For thesuccessive \(n\), the metric takes the form: \[dl_{\left( 0\right) }^{2}=\frac{1}{2}dU^{2}+\frac{3}{8}dV^{2},\]\[dl_{\left( 1\right) }^{2}=-\frac{1}{2}dU^{2}+\frac{5}{8}dV^{2},\]\[dl_{\left( 2\right) }^{2}=-\frac{5}{2}dU^{2}-\frac{13}{8}dV^{2}.\]It can be seen from this that for \(n\geq 2\) the distance becomes purely imaginary.

6 Entropy as the measure of a distance in Hilbert space↩︎

A measure of a "distance" between states \(\widehat{\rho }\) and \(\widehat{\sigma }\) is the relative entropy \(S_{rel}\) :\[S_{rel}\left( \widehat{\rho }||\widehat{\sigma }\right) =Tr\left[ \widehat{\rho }\log \widehat{\rho }-\widehat{\rho }\log \widehat{\sigma }\right] .\] It can be rewritten as follows\[S_{rel}\left( \widehat{\rho }||\widehat{\sigma }\right) =S\left[ \widehat{\sigma }\right] -S\left[ \widehat{\rho }\right] +Tr\left[ \left( \widehat{\sigma }-\widehat{\rho }\right) \log \widehat{\sigma }\right] , \]where:\[S\left[ \widehat{\sigma }\right] =-Tr\left( \widehat{\sigma }\log \widehat{\sigma }\right)\]is von Neumana entropy. The entropy \(S_{rel}\) is positive : \(S_{rel}\left( \widehat{\rho }||\widehat{\sigma }\right) \geq 0\) and is equal to zero if \(\widehat{\rho }=\widehat{\sigma }\). Thus one can consider this entropy as the as the square of the "distance" between states [18, p.44]. We compute this distance for pure states. Let the pure states have the form: \(\widehat{\sigma }=diag(0,..,1,0,..,0)\) (\(1\) is in \(a-th\) place) and \(\widehat{\rho }=diag(0,..,1,0,..,0)\) (\(1\) is in \(b-th\) place). Since that \(\left( I-\widehat{\sigma }\right) ^{n}=I-\widehat{\sigma }\) and \[\ln \left( \widehat{\sigma }\right) =\ln \left[ I-\left( I-\widehat{\sigma }\right) \right] =-\sum_{n=1}\frac{\left( I-\widehat{\sigma }\right) ^{n}}{n}\]one obtains that \(S_{rel}\left( \widehat{\rho }||\widehat{\sigma }\right)\) is equal to:\[S_{rel}\left( \widehat{\rho }||\widehat{\sigma }\right) =-Tr\left[ \left( \widehat{\sigma }-\widehat{\rho }\right) \left( I-\widehat{\sigma }\right) \right] \sum_{n=1}\frac{1}{n}. \]Using relations: \(\widehat{\sigma }=\widehat{\sigma }^{2}\) and \(Tr\left( \widehat{\sigma }\widehat{\rho }\right) =0\) finally we get:\[S_{rel}\left( \widehat{\rho }||\widehat{\sigma }\right) =Tr\left[ \widehat{\rho }\right] \sum_{n=1}\frac{1}{n}=+\infty . \]Hence the square of the "distance" between two pure states is infinite! In the case when \(\widehat{\sigma }\) is pure state and \(\widehat{\rho }\) is mixed state then:\[\begin{gather} S_{rel}\left( \widehat{\rho }||\widehat{\sigma }\right) =-S\left[ \widehat{\rho }\right] -Tr\left[ \widehat{\rho }\log \widehat{\sigma }\right] = \notag \\ \left( 1-Tr\widehat{\rho }\widehat{\sigma }\right) \sum_{n=1}\frac{1}{n}-S\left[ \widehat{\rho }\right] . \end{gather}\]However in this case this square of the "distance" can be finite, because the difference of the infinity sum of the harmonic series and entropy of the mixed state (which can also be infinity) could give finite results.

6.1 Thermal states and relative entropy↩︎

The thermal state is defined by maximum entropy \(S\) and fixed energy \(E\). Thus for an energy operator \(\widehat{H}\) a density matrix \(\widehat{\sigma }\) is equal to:\[\widehat{\sigma }_{t}\left( \beta \right) =\frac{1}{Z}\exp \left( -\beta \widehat{H}\right) , \]where \(Z=Tr\left[ \exp \left( -\beta \widehat{H}\right) \right]\) and \(\beta =1/\left( k_{B}T\right)\) is the Lagrange multiplier, which has interpretations of the inverse of temperature \(T\). The relative entropy \(S_{rel}\) between the states \(\widehat{\sigma }_{t}\) and \(\widehat{\rho }\) takes the form:\[S_{rel}\left( \widehat{\rho }||\widehat{\sigma }_{t}\right) =\beta Tr\left( \widehat{\rho }\widehat{H}\right) -\beta Tr\left( \widehat{\sigma }_{t}\widehat{H}\right) -\left( S\left[ \widehat{\rho }\right] -S\left[ \widehat{\sigma }_{t};\beta \right] \right) , \]where \(S\left[ \widehat{\sigma }_{t};\beta \right]\) is maximum entropy \(S\) and the term \(Tr\left( \widehat{\sigma }\widehat{_{t}H}\right)\) is fixed energy \(E\):\[Tr\left( \widehat{\sigma }_{t}\widehat{H}\right) =E\left( \beta \right) .\] Hence:\[S_{rel}\left( \widehat{\rho }||\widehat{\sigma }_{t}\right) =\beta Tr\left( \widehat{\rho }\widehat{H}\right) -S\left[ \widehat{\rho }\right] +S\left[ \widehat{\sigma }_{t};\beta \right] -\beta E\left( \beta \right) . \]The last two terms are related to free energy: \(F\left( \beta \right) =E-S/\beta\). Finally the relative entropy is equal to:\[S_{rel}\left( \widehat{\rho }||\widehat{\sigma }_{t}\right) =\beta Tr\left( \widehat{\rho }\widehat{H}\right) -S\left[ \widehat{\rho }\right] -\beta F\left( \beta \right) . \]

If also \(\widehat{\rho }\) is thermal state with an energy operator \(\widehat{h}\)\[\widehat{\rho }_{t}\left( b\right) =\frac{1}{U}\exp \left( -b\widehat{h}\right) , \][where \(U=Tr\exp \left( -b\widehat{h}\right)\)], then their relative entropy is:\[S_{rel}\left( \widehat{\rho }_{t}||\widehat{\sigma }\right) =-S\left[ \widehat{\rho }_{t};b\right] -\frac{1}{U}Tr\left[ \exp \left( -b\widehat{h}\right) \log \widehat{\sigma }\right] . \]The other form of the last equation is:\[S_{rel}\left( \widehat{\rho }_{t}||\widehat{\sigma }_{t}\right) =S\left[ \widehat{\sigma }_{t};\beta \right] -S\left[ \widehat{\rho }_{t};b\right] -\beta Tr\left( \widehat{\sigma }\widehat{H}\right) +\frac{\beta }{U}Tr\left[ \exp \left( -b\widehat{h}\right) \widehat{H}\right] . \]In this way the relative entropy for two thermal states with the fixed energy operators \(\widehat{H}\) and \(\widehat{h}\) is parametrized by the two positive numbers \(\beta\) and \(b\), with an interpretation of the inverse temperatures (modulo Boltzman constant).

When \(\widehat{H}=\widehat{h}\), then the last relation becomes:\[S_{rel}\left( \widehat{\rho }_{t}||\widehat{\sigma }_{t}\right) =S\left[ \widehat{\sigma }_{t};\beta \right] -S\left[ \widehat{\rho }_{t};b\right] +\beta E\left( b\right) -\beta E\left( \beta \right) . \]

As an example, we will find the relative entropy for a free scalar field in thermodynamic equilibrium. Such a field is equivalent to an infinite set of non-interacting harmonic oscillators with maximum entropy\(S\) and fixed energy \(E\). The entropy is equal to:\[S\left[ \widehat{\sigma }_{t};\beta \right] =2^{4}\frac{\pi ^{5}k_{B}}{45}\frac{1}{\left( \hbar c\right) ^{3}}\frac{V}{\beta ^{3}}, \]and the energy \(E\left( \beta \right)\) is equal to:\[E\left( \beta \right) =\frac{4\pi \cdot 3!}{\left( \hbar c\right) ^{3}\beta ^{4}}V\frac{\pi ^{4}}{90}+E_{0} \]where \(E_{0}\) is the infinite ground-state energy and \(V\) is volume of space. Hence the relative entropy takes the form:\[S_{rel}\left( \widehat{\rho }_{t}||\widehat{\sigma }_{t}\right) =\frac{4\pi ^{5}Vk_{B}}{45\left( \hbar c\right) ^{3}\beta ^{3}}\left[ 1-4\left( \frac{\beta }{b}\right) ^{3}+3\left( \frac{\beta }{b}\right) ^{4}\right] .\] If the parameter \(\beta\) is in the vicinity of \(b\):\[\beta =b+\delta \]for \(0<\delta /b<<1\), then their relative entropy is equal to:\[S_{rel}\left( \widehat{\rho }_{t}\left( \delta \right) ||\widehat{\rho }_{t}\left( b+\delta \right) \right) =\frac{4\pi ^{5}Vk_{B}}{45\left( \hbar c\right) ^{3}}6\frac{\delta ^{2}}{b^{5}}+O\left( \delta ^{3}\right) .\] Taking the interpretation of relative entropy as the square of the distance between states, the above relationship can be written as an expression for a one-dimensional (one parameter\(b\)) metric:\[dl^{2}=A\frac{db^{2}}{b^{5}}, \]where \(A=8\pi ^{5}Vk_{B}/\left[ 15\left( \hbar c\right) ^{3}\right]\). In term of the energy \(E\) the last formula takes the form:\[dl^{2}=\frac{k_{b}}{8}\left[ \frac{4\pi ^{5}V}{15\left( \hbar c\right) ^{3}}\right] ^{1/4}E^{-5/4}dE^{2}. \]So the distance in this metric between two states of a free scalar field with fixed energies \(E_{2}>E_{1}\) is equal to:\[d\left( E_{1},E_{0}\right) =\frac{1}{3}\sqrt{8k_{B}}\left[ \frac{4\pi ^{5}V}{15\left( \hbar c\right) ^{3}}\right] ^{1/8}\left( E_{2}^{3/8}-E_{1}^{3/8}\right) . \]

7 Conclusion↩︎

The method of determining metric and distance in classical and quantum physics is shown in the diagram below. The left column refers to classical physics and the right column refers to quantum physics: \[\begin{array}{ccc} \left( E,\mathbf{p}\right) & & \left( \widehat{\rho },?\right) \\ \downarrow & & \downarrow \\ \left( T_{\mu \nu }\right) & & \begin{array}{c} \left( S,?\right) \\ \downarrow\end{array} \\ \begin{array}{c} \begin{array}{c} \downarrow \\ \left( \left( T_{\mu \nu }\right) ,\left( R_{\nu \rho \sigma }^{\mu }\right) \right) \\ \downarrow (\text{EP)}\end{array} \\ \delta \int \left( R+\Lambda +L_{m}\right) =0 \\ \downarrow \\ d=d\left( x,y;\left( g_{\mu \nu }\right) \right)\end{array} & \begin{array}{c} \Longleftarrow \\ ?\end{array} & \begin{array}{c} \mathcal{H} \\ \downarrow \\ \widehat{H}\psi =i\partial _{t}\psi \\ \downarrow \\ d\left( \widehat{\rho },\widehat{\sigma }\right) =...\end{array}\end{array},\]where (EP) means equivalence principle. As it follows, EP is the key idea that is needed to determine the geometry of spacetime. Question marks indicate unknown complementary concepts. It can be said that an ambiguity in determining the distance or metric in quantum physics (e.g [19-21]) is due to the lack of an counterpart of the equivalence principle.

Moreover there is an intriguing relationship between pure states and the Kasner metric in \(n+1\) dimensional spacetime. This metric describes vacuum cosmological solutions and has the form:\[ds^{2}=dt^{2}-\dsum\limits_{a=1}^{n}t^{2p_{a}}dx_{a}^{2},\]where the numbers \(p_{1},...,p_{n}\) obey two constraints:\[\dsum\limits_{a=1}^{n}p_{a}=\dsum\limits_{a=1}^{n}p_{a}^{2}=1.\]Thus, formally, one can consider these numbers \(\left\{ p_{a}\right\}\) as the eigenvalues of a certain density matrix \(\widehat{\rho }\) corresponding to pure state.

The obtained metric (4.10a-b) on infinitly dimensional sphere \(S^{\infty }\) is given by the integrals (4.11) and was obtained as the FR metric for the infinitly dimensional Hilbert space. It is known that when the Hilbert space is \(CP^{N}\), then the FR metric becomes the Fubini-Study metric. Thus the metric (4.10a-b) should be reduced to Fubini-Study in the case of finite dimensions. But it is not obvious. The sphere \(S^{\infty }\) should be modeled by some kind of limit of N-dimensional spheres \(S^{N}:\)\["\lim_{N\rightarrow \infty }"S^{N}=S^{\infty }.\]Moreover the metrics on \(S^{N}\) have to be invariant under the unitary grup \(U\left( N\right)\). For finite \(N\) the normalization condtion leads to odd value of \(N=2n+1\) so the sphere \(S^{2n+1}\) is represented as the homogenous manifold:\[\frac{U\left( n+1\right) }{U\left( n\right) }\simeq S^{2n+1}.\]From the other side \(CP^{n}\) is represented also as the homogenous manifold:\[\frac{S^{2n+1}}{S^{1}}\simeq CP^{n}.\]One of the \(U\left( n\right)\) invariant metric is Fubini-Study (FS) metric. But there are other metrics which remain hidden. In the FS metric the volume of \(CP^{N}\) is equal to:\[vol_{FS}\left( CP^{N}\right) =\frac{\pi ^{N}}{N!}.\]Hence for \(N\rightarrow \infty\) the volume in the FS metric tends to zero. It can be seen from here that FS metric do not lead to the metric (4.10a-b).

In the case of a measure related to relative entropy, completely different distances are obtained [e.g.21-25] , which should not be surprising. Without some fundamental principle each measure is equally appropriate. In this article, we have presented only very specific measures. Of course, there are more of them.

One of the unsolved problems in quantum cosmology is finding the "norm" of the universe’s wave function. In this case, the WDW equation is defined on a minisuperspace and the wave function is known for different boundary conditions [26-33]. So the use of the approach outlined in this work is most reasonable. Finding the "distance" for the wave functions describing the expanding universe and the collapsing universe is a task to be accomplished. A more difficult issue is the "distance" between states associated with black holes. These issues will be addressed in the next paper.

Author Contributions: Conceptualization, PG.; methodology, PG, DB, AR. All authors have read and agreed to the published version of the manuscript.

Funding: This research received no external funding.

Data Availability Statement: Data are contained within the article.

Conflicts of Interest: The authors declare no conflicts of interest.

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