Bohm's Quantum Mechanics

Note this is rough version 0.3 which has not yet been corrected for technical errors, wrong placement of equations etc. Future versions will also have hyperlinks explaining advanced terms. The Editors

Let us write the Schrodinger equation as

where H is the Hamiltonian. Then the differential of the wave function is

Now the wave function lives in a Hilbert space, H, and the projective Hilbert space is PH. The elements in the projective Hilbert space are related to those in the Hilbert space by

then the above differential becomes

Now multiply through by psi^dag and we are left with

The first term on the right had side gives the geometric phase of M. Berry and the second is the dynamical phase [1]. Once integrated over a closed path the integral int_C dy, gives the first Chern class of the manifold of dynamics [2,3]

Topological Numbers and Quantum Vortices

If one wants to look for topological numbers associated with the quantum force or the amplitude I think the best way is to use differential forms. This means that one should write for a wave function written as psi~=~rho^{1/2} e^{iS} that,

where C is a closed loop the system traverses. Now Stokes' law gives the result

where C is a closed curve and A is any area bounded by this curve. The two-form may be expanded and since rho~=~rho(x,t) the two-form (d rho) v (d rho), v is used a the wedge product, is a product of gradients multiplied by the two-form dx^j \wedge dx^i. The total time derivative of the action is:

The Hamilton-Jacobi equation &S/&t~=~-H, & = partial, may be used to write the differential of the action as

and an expansion of the differential of the Hamiltonian in the basis of forms dx^i v dp^j. Our loop integral is then

The second equation contains the commutator between the density and the classical Hamiltonian. The other two terms are the real valued portions, while the second is the imaginary portion. This equation is a differential -commutator equivalent forms of the usual decomposition of the Schr{\"o}dinger equation. The expectation of this loop integral

is the Berry phase.

To those familiar with the particle view of quantum mechanics the real part to the Schr{\"o}dinger equation is a modified Hamilton Jacobi equation and the imaginary part is a continuity equation for the flow of a "quantum hydrodynamic fluid." Physically this means that if [&_j rho,~ &_i rho] is nonvanishing, then the density has a rotational component to it. The quantum hydrodynamic fluid, or pilot wave, has a vortex. The value of this integral is a measure of the singularity or pole at the center of the whirlpool.

It has been indicated that this is similar to Feynman's vortex theory of super fluids, Helium II[4]. This is a reasonable observation. The Feynman rotons are in effect quantum wave functions on a large scale. I will return to that in a minute. Another way of looking at this is according to interference experiments. Say we have a two slit experiment. Now the sum over experiments gets its statistics from the superposition of the two paths c_1 and c_2 that connect the source and a detected spot on the photoplate through the two slits. We can equally view this as a loop that psi~=~rho^{1/2}e^{iS},~ hbar~=~1, sends a particle through.

Now what does all of this mean? Let's go back to the Schr{\"o}dinger equation

which tells us the change in psi due to an increment in time. What our expression is telling us is that if we think of the particle as having traversed the loop between the source gun to the screen and back to the source gun that there is an increment in time, or delta time, between the two halves of the loop, weighted by the eigenspectrum. Another view is that this loop integral measures the interval of time as a path length for the parallel translate of rho^{1/2}drho +idS around the loop. If the interval of time for c_1~=~c_2 then the entire delta t for traversal is zero and the value of our integral is zero, and this is destructive interference. Otherwise, where the integral is nonzero there is a nonzero measure for the delta t associated with the particle traveling around the loop. I think this is a beginning to understand superposition according to the particle view of QM. The superposition principle can be looked at as a Gauss-Bonnet theorem for the delta t a particle takes to traverse a loop.

Ok back to the idea of superfluidity. This will be a heuristic introduction to what is to follow. Let the differential operator be

Then our commutator of the density goes as

where the additional part are the field components of a Yang-Mills gauge theory. If we are in a U(1) domain then the last commutator vanishes and we have the magnetic field that satisfy Maxwell's equations. This is completely equivalent to a Berry phase that is induced by an electromagnetic field. The commutator describes the vortex motion of the qauntum hydrodynamic fluid, or pilot wave, for the system. It has strong analogs to Feynman's rotons in liquid helium II [5].

Density Operator Methods

Let

be the density operator that transforms according to the unitary group

The evolution equation for the density operator is then

The time derivative of the generator of the unitary transform is

where for

This may be expressed as the loop integration

By Stokes theorem the second loop integral is

and under the above transformation and a component by component breakdown of the commutator is

Now let the time dependent generator of the time development operator according to a time dependent Hamiltonian term,

so that

Now use the parameterization of this time dependent Hamiltonian according to the position and momentum z~=~z(t), ~z~=~{x,~p},

Then this two-form can be written according to the Poisson bracket

Now the differential of the density matrix evaluated on a closed loop is

The intended us of this is to consider Hamiltonians of the form

where omega_i are the angular frequencies. Then an examination of a punctured KAM and Cantori should give information of the extent to which the quantum hydrodynamic fluid diverges by the ensuing chaos or turbulence. As such the above time loops become unpredictable for the particle as it wildly dances in the Cantor dust of the shattered KAM surface.

At this point we must return to some analysis. Since the time dependent Hamiltonians evolve according to above unitary operator,

and so the differential two-form is,

Now evaluate this two form under a trace, or sum over states

Now set t~-~t'~=~tau and this integral becomes

To conclude, this part our loop integral over the density matrix is now

Since we are working with Hamiltonian systems the space phase volume of the system is conserved. The surface of this volume in phase space is determined by the conservation of energy, and defines the energy surface of the system,

where @ (@ = theta) is the Heavyside function. The energy defines the boundary of this volume in phase space,

The trace over any operator {overhat O} is defined on the energy surface where expectation values exist,

Now for overhat O~=~d{tilde H}_{tau} v d{tilde H}_0 we have

Now use the algebra of commutators to rewrite this as

The first commutator is a surface term and thus vanishes. The second commutator is by the chain rule

Now the Poisson bracket here defines the time evolution of the Hamiltonian {tilde H}_{tau}. Further, the derivative of the delta function when evaluated on a function just returns the derivative of that function at the stationary point. This then leads to for {overhat O}

This is a fundamental equation, but some simplifications are required, since the appearance of t' is troubling. To eliminate this problem let us introduce the limit function lim_{epsilon ---> 0}e^{-epsilon t} into the integral. For epsilon ----> 0 it is a simple matter to show that

and so our above result can be expressed in the more aesthetic form

Now if this is evaluated along the loop integration we have

Hamiltonian Chaotic Systems

Now let us consider chaotic systems. Let the time dependent Hamiltonian have the Fourier expansion

where we will work within the action angle variables J, @ so that

Now set @(t)~=~@_0~+~omega(I)t, with d@/dt~=~grad_J H_0. Then we have the two-form

The gradient along the action variables of this equation is then

Now we have grad_J H_n = J_n, and so this reduces to

where the last equality absorbs n into J_n. The term J_n*J is now a product between two different action variables. We could just as well write this as

where g_{ij} is a metric for the energy surface.

is a curvature two-form evaluated on an enclosed area of the energy surface. Further, the curvature is proportional to the metric. This means that the space is an Einstein space, with symplectic coordinates.

Here is where chaos enters the picture. A dynamical system is chaotic if any two initial points arbitrarily close become widely separated within a finite period of time. Let the distance between any two points on the energy surface be given by a line element,

Then the separation of any two points is given by

The Liapunov exponent for a dynamical system is defined as

For a chaotic system this distance will separate at an exponential rate and the Liapunov exponent is nonzero. The tori of regular dynamics have elliptical curvatures, which means that any two intersecting geodesics are guaranteed to recross at some other point on the space. A manifold with a hyperbolic curvature will have divergent geodesics that will exponentially separate with time. The Gaussian curvature of a hyperbolic manifold is negative, and is a reasonable geometry to use for the study of chaotic systems. The break down of tori leads to regions of stochastic behavior. The positive Gaussian curvature of the torus is being made negative in regions of overlap across KAM surfaces of separation.

An easy example to demonstrate are the geodesics on the Poincare half plane. This two-dimensional space has the line element

The connection coefficients are,

The geodesic equations are

The solutions to this equations are x = alpha + beta tanh(t) and x = beta sech(t). The distance between two points y_1 and y_2 diverge as

diverges as y_1 ----> 0 and y_2 ----> infinity, or equivalently as $T ----> infinity. Further, the Gaussian curvature is,

This space is a resonable toy that captures the hyperbolic structure of the geometry that occurs with the breakdown of a KAM surface.

Quantum Geometry and Bohm's Theory

Now let us examine this within the context of Bohm's particle plus pilot wave model of quantum mechanics. When the wave function is written in a polar form

the Schrodinger equation splits into a real and imaginary parts[6],

which are the quantum corrected Hamilton-Jacobi equation and continuity equation respectively. Now the particle's momentum is p = grad S, which defines the Lagrangian manifold of constant action in phase space. Now let the momentum operator {overhat p} = {hbar/i}grad act on the wave function psi = rho^{1/2}e^{iS} = e^{R~+~iS}

The expectation value of the momentum operator is

since grad S = p this is written in the more compact form

This must mean that i = 0. Now put 1 = tr rho in the right hand side of this equation

Now just peal one level of < > off and we are left with

The particle's momentum is the expected value plus a quantum correction to this momentum,

The particle's momentum is then the expectation p =

plus the fluctuation,

It is easily see that = 0. Now define beta^{+-} = grad(R +- iS). Now the two equations become

Bohm's quantum potential is then equal to the square of the quantum fluctuations plus a modified quantum potential Q' = {hbar/2}grad^2 R

The equations of motion are found by taking the gradient of the modified Hamilton-Jacobi equation:

Now use the fact that

to find that the equation of motion is,

Similarly the gradient on the continuity equation gives the fluctuation in the force as,

Now suppose that beta^{+-} depends upon some internal set of coordinates that transform according to a Lie group G, beta^{+-} = beta_a^{+-} e^a. Then the generalized velocities beta_i^{+-} become

where {A^a}_{bj} is a component of a gauge connection and g is a coupling constant for a Yang-Mills gauge field. It is then straight forward to calculate the action of commuting differential operators on beta^{+-},

These gauge field equations may be substituted into the equations of motion:

and

We may make these equations of motion more symmetrical by finding &(p_i +- i pi/&t. Further, the internal symmetries are allowed to have time dependencies. Then we are left with the coupled set of differential equations,

Physically the gauge field terms have two meanings. The terms beta^{j +-}F_{ij} are Lorentz forces. The field strength terms F_{ij} are the magnetic field components of the gauge theory epsilon_{ijk}B^k = F_{ij} and the force is in a direction orthogonal to the generalized velocity v_j + delta v_j = beta_j/m. The second term partial^j F_{ij} are the currents, which are zero in a source free region. The terms &A_i/&t are the electric field components.

At this point it should be pointed out that the gauge theory is seen to emerge from the quantum potential

when beta^{+-} = R +- iS is dependent on internal degrees of freedom governed by a Lie algebra. This is different from the usual view where the internal degrees of freedom are assigned to each point on the base manifold. The electric field components of the gauge theory are found in the potential in the Hamilton-Jacobi equation, V = V_0 + phi, and E_i = &_i\phi. For the case where the gauge theory is the electromagnetic field the abelian group structure is reflected in the vanishing of the commutator between the gauge connection.

Before we examine the geometry of a quantum system let us touch base with some basic notions of quantum mechanics. We have from the Bohm approach to quantum theory that the momentum of a particle is given by a classical part plus a fluctuation. By construction we have that = 0 and

The Heisenberg equations of motion assume the form

The potential may be expanded in terms of , and then the fluctuations obey the equations of motion

The Lyapunov exponent used to measure the sensitivity on initial conditions of a dynamical system. The exponential deviation between the positions of two particles in phase space generate this quantity,

A nonzero value for L indicates that the trajectories for two different particles whose initial positions in phase space differ by a fluctuation (delta x(0), delta p(0)). Then if the fluctuation is propagated exponentially with time, then the system is chaotic. For classical systems this fluctuation represents a small deviation in the trajectories between two particles, or an uncertainty in the position of a single particle. This uncertainty exponentially propagates for a chaotic system. For a quantum particle this uncertainty is due to the stochastic fluctuations due to the quantum force on the particle.

The fluctuations are found from the equations of motion

The potential may be expanded in a Taylor series around in powers of the fluctuation delta x. Yet for a chaotic system the fluctuations grow exponentially and the Taylor series can not be satisfactorily truncated. As a result the second expression may be more useful, but difficult to apply.

Now let us examine the loop integral oint psi^{dag}psi for psi = e^{beta/hbar},

The differential of the exponentiated term in the wave function may be written as

With some analysis and the definition H^{+-} = H_r +- iH_s the components of the two-form that projects through the area of integration is a Poisson bracket

The Poisson bracket is evaluated with the Hamilton's equation for the position and momentum and their fluctuations,

Use of these Hamilton's equations and the dynamical equations of motion lead to the two-form,

where

The two-form d beta^- v d beta^+ is evaluated on a parallelogram on the energy surface of the system. This energy surface is the same as computed according to the Bohr interpretation, but where the "fuzziness" of quantum fluctuations pi v pi and pi v p are explicitly included. The energy surface does not have the same sharpness given by a Heavyside function. The computation on the energy surface according to the Bohr approach is an expectation of the energy surface just computed according to the Bohm method. As such the expected energy surface is one where = 0$ has been taken, and the quantum input from the expectation over the Bohm energy surface is due to terms of the form =/ 0 that emerge from the quantum potential.

The same analysis on the time integration of the Poisson bracket {H^{+}(t'), H^{-}(t")} may be performed. Without need of weighty analysis the result shall just be stated:

If we write these generalized Hamiltonians as H^{+-} = sum_n H_n^{+-} e^{-+ i @(t)} we arrive at a result that has the form

where J^- = grad H^-(0) and J_n^+ = grad H_n^+ These action variables by construction contain quantum fluctuations, J^{+-} = J +- i delta J. With this the two form is

where the terms linear in the fluctuations will have expectations that vanish.

For a chaotic system the Lyapunov function will involve the fluctuations. The fluctuation in the action variables will have their own contribution to the Liapunov exponent,

which is nonzero for a chaotic system. Then a fluctuation will proceed to grow with time. This is contrary to the usual notion of quantum fluctuations that are Markovian. Markovian fluctuations are such that the fluctuations in a system at one time are independent of those at any other time. Any transition between states, |i> ----> |j> through a fluctuation, such as tunneling, at a time t' occur through an interaction that is of the form (i|V|j) = omega_{ij}delta(t - t'). For a regular dynamical system the geometry of the energy surface is elliptic, and so any fluctuation the results in a deviation of the system will in a finite period of time intersect a path that would have resulted from another fluctuation. If the energy surface is hyperbolic then the fluctuation will be amplified according to exp(L(t)). This means that the dynamics of the particle, according to Bohm, will be highly dependent upon the initial configuration of the amplitude, or equivalently on the value of the quantum potential.

Since the quantum force can be written as

and that the wave packet oscillations are due to the behavior of the quantum potential we can write the wave function as

where F is a periodic function in time. Then given any observable {overhat O} its expectation is

This construction means that the Lagrangian manifold for the classical system is associated with a foliation of Lagrangian submanifold

This foliation of the Lagrangian surface is seen in the fuzziness of the energy surface due to the term i(J*delta J_n - \delta J*J_n) in the two form beta^* v beta. For a chaotic system the value of these fluctuations grows exponentially. The observable role of this growth is unmeasurable as , but is so as <(\delta p)^2>.

Discussion, Problems, and Future Issues

What we have is a demonstration of how fluctuations derived from Bohm's approach to quantum mechanics can be associated with a Lyapunov exponent for a Hamiltonian chaotic system. This fluctuation can involve both the momentum fluctuation of the particle and the associated gauge fields derived when the amplitude has a set of internal degrees of freedom. This promotion of a fluctuation in a quantum system goes against the often stated notion that quantum fluctuations are delta function correlated so that fluctuations at one time are independent of those at any other time.

The gauge fields derived from the methods just presented are purely magnetic in nature. They could easily be magnetic fields, or the magnetic analog in QCD. For most practical issues this theory is designed for the electromagnetic interaction is likely to be used. The theory has so far an "undemocratic" treatment of the gauge field, where the electric field part in this theory does not contribute a source term grad*E = rho, where as the magnetic field partion of the gauge field does with the terms grad^j F_{ij} and its hermitian complement. For a source free region this distinction should present no problem. Indeed, for a problem where there are no static free charges this should still not be a problem. For a static charge to exist within this theory there should be a term

again for beta^{+-} = R +- iS. This term unfortuanately does not exist. For the issues of quantum biophysics this is a difficulty. Most processes in cellular biology involve potential differences across cell membranes. The ligand gates of neurons and the ion pumps that interface the mitochondria and a cellular cytoplasm involve the transport of ions and potential differences across cellular membranes. The storage of charge across a cell membranes is a capcitive part of an driven RC circuit equation for the propagation of an action potential between two neurons.

Yet this problem is solved if the quantum potential is explicitely time dependent,

Q will then contain at second order in the fluctuation, delta x delta t, Q will contain a term that has a term

multiplied by delta xdelta t in the expansion. The term should then be observable since |delta()| > 0. This is an issue to be exploered more fully. It is curious that the static source term that would contribute to quantum biophysics is on such a low order. The contribution of currents is quite small with biophysics, as is seen in the requirement to use Josephson junctions to measure quantum magnetic flux quanta associated with neurophysiological measurements. Yet this might mean that the contribution of currents and stored charge are equivalent on the quantum level. Much of the classical associated with neurophysicalogical processes is large, but at the quantum level the magnetic and electric contributions are equivalent. This is a regime that needs serious theoretical and experimental exploration.

1) M. V. Berry, Proc. Roy. Soc. London, {\bf a392}, 45, (1984)

2) D. A. Page, Phys. Rev. A, {\bf36}, 3479, (1987)

3) J. Sarfatti, Private Communication

4) S. S. Chern, (editor and Author of Paper),{\it Global Differential Geometry}, Wiley (1991)

5) R. P. Feynman, phys. Rev., {\bf 91}, 1291, (1953).

6) D. Bohm, Phys. Rev.,{\bf 85}, 166, (1952).