Chattaraj P.K. (Ed.) Quantum Trajectories

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74 Quantum Trajectories

spacetime. Our purpose here is to show how the scheme may be extended to include

the derivation of the dynamics of quantum ﬁelds from a trajectory model. It turns out

that a uniﬁed description embracing both bosons and fermions may be developed by

generalizing the conﬁguration space of the known scheme for ﬁrst quantized systems

to a Riemannian manifold. We shall illustrate this for the simplest non-trivial systems

that exhibit the key algebraic features, namely, non-relativistic free spinless boson

and fermion ﬁelds. In the ﬂuid description the two types of statistics are distinguished

by the choice of the conﬁguration space and associated Riemannian metric, together

with the potentials to which the ﬂow is subject. Speciﬁcally, the trajectories in the case

of bosons will be described by translational time-dependent oscillator coordinates,

and an analogous system of rotational time-dependent coordinates is employed in the

fermion case. We thus call attention to the value of considering ﬂows in spaces other

than physical (Euclidean) space.

In extending the continuum-mechanical construction of the wave equation to

fermionic ﬁelds we shall exploit the fact that the anticommutative fermion algebra is

essentially an algebra of SU(2) angular momentum operators. It is necessary then to

include spin

in the hydrodynamical description. This problem has been addressed

previously and indeed the inclusion of spin was the initial motivation for develop-

ing the Riemannian generalization that we use [4, 5]. In presenting this method here

we generalize the previous treatment slightly to include external potentials, which

are necessary for the application to both boson and fermion ﬁelds. The key point in

the fermion case is that in the Lagrangian picture the ﬂuid particles are treated as

spin

rotators described by sets of Euler angles. The necessity of this model arises

because the usual textbook approach to spin employing spinors is not suitable for our

purposes. We shall discuss this problem ﬁrst.

5.2 SPIN AND TRAJECTORIES

An Eulerian hydrodynamic picture for spin

follows in a fairly straightforward way

for systems governed by the Pauli equation (e.g., Ref. [6] and references therein).

However, the hydrodynamic equations in this approach to spin are rather complicated

and difﬁcult to interpret, at least in comparison with the simplicity of the spin 0

Madelung model, and there are ambiguities in identifying suitably deﬁned phases of

the spinor components. An additional problem in connection with our constructive

program is that it is not clear how to develop a suitable Lagrangian-coordinate, or

trajectory, version of the theory. We may see this with a simple example. For a spinor

ﬁeld ψ

(x), a = 1, 2, the associated probability density is

ρ = ψ

†

ψ (5.1)

and the ﬂow lines are naturally deﬁned as the integral curves of the velocity ﬁeld

(x) =



2miρ



†

∂ψ

∂x

−

∂ψ

†

∂x



, i = 1, 2, 3. (5.2)

Undoubtedly, these paths provide considerable insight into spin phenomena [6]. The

problem in the present context is that the ensemble of paths generated by varying

Quantum Field Dynamics from Trajectories 75

the initial position does not contain sufﬁcient information to construct the four real

components of the spinor ﬁeld according to the method set out above and described

in [2]. To see the nature of the information that is missing, consider the following two

wavefunctions (ψ



obeys the free Pauli equation if ψ does):

ψ(x, t ) =





, ψ



(x, t ) = σ

ψ(x, t ) =





(5.3)

where we have used one of the Pauli matrices σ

, i = 1, 2, 3 (any one would do

in this example). These wavefunctions may be chosen to have non-trivially distinct

time dependences. But the implied probability density and velocity are the same

for each, given by the Equations 5.1 and 5.2. To capture the further information in

the wavefunction not exhibited by the trajectories one needs to introduce additional

hydrodynamic-like variables built from the spinor. The spin vector ﬁeld s

(x, t ) =

(/2ρ)ψ

†

ψ plays such a role; denoting by s



the spin vector associated with ψ



, the

two states (Equation 5.3) are distinguished as follows:



= s

, s



=−s

, s



=−s

. (5.4)

Unfortunately, while such quantities suitably extend the Eulerian hydrodynamic des-

cription, this does not assist us since the additional functions do not have Lagrangian

counterparts. That is, although one can potentially associate additional structure with

the trajectories deﬁned by Equation 5.2 by evaluating the functions along them, the

trajectories exhibit no features that could be set in correspondence with the functions.

One can partially ameliorate the situation by seeking to redeﬁne the velocity so that

it depends on the additional variables whilst corresponding to the same density. Just

such an additional term is, in fact, required by relativistic considerations [7] and this

leads to the following modiﬁed non-relativistic velocity:



2miρ



†

∂ψ

∂x

−

∂ψ

†

∂x



mρ



j,l

ijl

∂

∂x

(ρs

). (5.5)

It remains the case, however, that we cannot uniquely infer from the modiﬁed trajec-

tories the full time-dependent wavefunction.

The problem is compounded for many-body systems. As the number of particles

increases, reproducing all the information in the wavefunction through additional

hydrodynamic variables requires ever more complex quantities that lack physical

interpretation and corresponding Lagrangian representatives.

It has been argued that the origin of the problems with the spin

hydrodynamic the-

ory in this context—its complexity and lack of an adequate Lagrangian formulation—

is that the standard approach to spin on which it is based works with a (angular

momentum) representation of the quantum theory in which the rotational freedoms

appear as discrete indices in the wavefunction [6]. The local ﬂuid quantities (such

as density and velocity) are deﬁned by summing, or “averaging,” over these indices,

and as a result essential information contained in the wavefunction is lost. The alter-

native procedure advocated in Reference [6] is to start from the angular coordinate

76 Quantum Trajectories

representation in which the spin freedoms are represented as continuous parameters

(α

, α

) (Euler angles) in the wavefunction, on the same footing as the spatial

variables x: ψ(x, α, t). Speciﬁcally, we write

ψ(x, α, t) =



a=1,2

(x, t )u

(α), (5.6)

where

(α) = (2

√

2π)

−1

cos(α

/2)e

−i(α

+α

)/2

(α) =−i(2

√

2π)

−1

sin(α

/2)e

i(α

−α

)/2

(5.7)

are spin

basis functions with



∗

(α)u

(α)dΩ = δ

dΩ = sin α

dα

, α

∈[0, π], α

∈[0, 2π], α

∈[0, 4π]. (5.8)

This implies a physically clearer and simpler hydrodynamic model. The phase S of

the wavefunction is immediately identiﬁable and the equations for the ﬂuid paths

are deﬁned in terms of the gradient of S with respect to all the coordinates, an obvi-

ous generalization of the spin 0 theory. The paths thus comprise both spatial and

angular components. The principal advantage of the angular coordinate approach for

our considerations here is that it provides a sufﬁciently detailed Lagrangian picture

that allows us to extend the constructive method to include the wavefunction (Equa-

tion 5.6). The enhanced detail is indicated, for example, by the fact that the velocity

ﬁeld (Equation 5.2) is the mean of the new velocity over the angles:

(x, t ) = (1/ρ)



|ψ(x, α, t)|

∂S(x, α, t)

∂x

dΩ. (5.9)

Having obtained the wavefunction, the components of the time-dependent spinor ﬁeld

may be extracted by using Equation 5.8 to invert Equation 5.6:

(x, t ) =



ψ(x, α, t)u

∗

(α)dΩ. (5.10)

In our application to fermions here the space variables are not relevant and we

need consider only the angular freedoms.

5.3 TRAJECTORY CONSTRUCTION OF THE WAVE EQUATION

IN A RIEMANNIAN MANIFOLD

5.3.1 L

AGRANGIAN PICTURE

We describe here the Lagrangian-coordinate hydrodynamic derivation of the wave

equation in an N-dimensional Riemannian manifold M with generalized coordinates

Quantum Field Dynamics from Trajectories 77

and (static) metric g

µv

(x), µ, v, ... = 1, ...,N . In this space, the history of

the ﬂuid is encoded in the positions ξ(ξ

, t) of the distinct ﬂuid elements at time t,

each particle being distinguished by its position ξ

at t = 0. We assume that the

mapping between these two sets of coordinates are single-valued and differentiable

with respect to ξ

and t to whatever order is necessary, and that the inverse mapping

(ξ, t) exists and has the same properties.

Let P

(ξ

) be the initial density of some continuously distributed quantity in M

(number of particles in our application) and g = detg

µv

. Then the quantity in an

elementary volume d

attached to the point ξ

is given by P

(ξ

)



|g(ξ

)|d

The conservation of this quantity in the course of the motion of the ﬂuid element is

expressed through the relation

P (ξ(ξ

, t))



|g(ξ(ξ

, t))|d

ξ(ξ

, t) = P

(ξ

)



|g(ξ

)|d

(5.11)

P (ξ

, t) = D

−1

(ξ

, t)P

(ξ

) (5.12)

where

D(ξ

, t) =



|g(ξ)|/|g(ξ

)|J (ξ

, t), 0 <D<∞, (5.13)

and J is the Jacobian of the transformation between the two sets of coordinates:

J =



,...,µ

,...,v

···µ

···v

∂ξ

···

∂ξ

. (5.14)

In our later application where we take the limit N →∞it is assumed that the

determinant converges.

We assume that the Lagrangian for the set of ﬂuid particles comprises a kinetic

term, an internal quantum potential energy that represents a certain kind of particle

interaction, and terms due to external scalar and vector potentials:

L =



(ξ

)





µ,v



µv

(ξ)

∂ξ

∂t

∂ξ

∂t

− g

µv

(ξ)



∂P

∂ξ

∂P

∂ξ





(ξ)

∂ξ

∂t

− V (ξ)





|g(ξ

)|d

. (5.15)

Here P

, A

, V, and g

µv

are prescribed functions, ξ = ξ(ξ

, t) and we substitute for

P from Equation 5.12 and write

∂

∂ξ

= J

−1



∂

∂ξ

(5.16)

where

∂J

∂(∂ξ

/∂ξ

)

(5.17)

78 Quantum Trajectories

is the cofactor of ∂ξ

/∂ξ

. The latter satisﬁes



∂ξ

= J δ

. (5.18)

It is assumed that P

and its derivatives take values such that the surface terms in the

variational principle vanish. The parameter m will later be identiﬁed with the mass

of the quantum system and should not be confused with the mass of a ﬂuid particle,

which is mP

√

|g|d

ξ.

Varying the coordinates, the Euler–Lagrange equations of motion for the ξ

th ﬂuid

particle take the form of Newton’s second law in general coordinates:

∂

∂t



v,σ



vσ

∂ξ

∂t

∂ξ

∂t

=−



µv

∂

∂ξ

(Q + V ) +



µv

∂ξ

∂t

(5.19)

where

vσ



µρ

(∂g

σρ

/∂ξ

+ ∂g

vρ

/∂ξ

− ∂g

vσ

/∂ξ

), F

µv

= ∂A

/∂ξ

−

∂A

/∂ξ

, and

Q =



µ,v

−

√

|g|P

∂

∂ξ





|g|g

µv

∂

√

∂ξ



(5.20)

is the quantum potential. Here we have written the force terms on the right-hand side

of Equation 5.19 in condensed form and substituting for P from Equation 5.12 and

for the derivatives with respect to ξ from Equation 5.16 we obtain a highly complex

fourth-order (in ξ

) local nonlinear partial differential equation. We shall see that from

the solutions ξ = ξ(ξ

, t), subject to speciﬁcation of ∂ξ

/∂t whose determination is

discussed next, we may derive solutions to Schrödinger’s equation.

5.3.2 I

NITIAL CONDITIONS

To obtain a ﬂow that is representative of Schrödinger evolution we choose the initial

conditions of Equation 5.19 so that P

is the square of the amplitude of the initial

wavefunction and the initial covariant components of the velocity ﬁeld are given by



µv

(ξ

)

∂ξ

∂t



∂S

(ξ

)

∂ξ

− A

(ξ

)



(5.21)

where S

(ξ

) is the phase of the initial wavefunction. To show that these assumptions

imply motion characteristic of quantum evolution we ﬁrst demonstrate that the form

Equation 5.21 is preserved by the dynamical Equation 5.19. To this end, we use a

generalization of the method employed in classical hydrodynamics based on Weber’s

transformation [8].

We ﬁrst multiply Equation 5.19 by



σµ

∂ξ

/∂ξ

and integrate between the time

limits (0, t). The term involving the Christoffel symbols {

vσ

}drops out and we obtain

Quantum Field Dynamics from Trajectories 79



σ,µ

σµ

(ξ(ξ

, t))

∂ξ

∂t



ρµ

(ξ

)

∂ξ

∂t

(ξ

) −



∂ξ

∂

∂ξ







µ,v

µv

(ξ(ξ

, t))

∂ξ

∂t

∂ξ

∂t



∂ξ

∂t

−

(Q + V )



dt. (5.22)

Then, substituting Equation 5.21, we get



σ,µ

σµ

∂ξ

∂t



∂S

∂ξ

−



∂ξ



(5.23)

where

S = S







µ,v

µv

∂ξ

∂t

∂ξ

∂t



∂ξ

∂t

− Q − V



dt. (5.24)

The left-hand side of Equation 5.23 gives the covariant velocity components at time t

with respect to the ξ

-coordinates. To obtain the ξ-components we multiply by J

−1

and use Equations 5.16 and 5.18 to get, at time t,



µv

∂ξ

∂t



∂S

∂ξ

− A

(ξ)



(5.25)

where S =S(ξ

(ξ, t), t). We conclude that the covariant velocity of each particle

retains forever the form Equation 5.21 if it possesses it at any moment.

5.3.3 E

ULERIAN PICTURE

We can transform the nonlinear dynamical equation into a linear one by changing

the independent variable from the particle label ξ

to a ﬁxed space point x. The link

between the particle (Lagrangian) and wave-mechanical (Eulerian) pictures is deﬁned

by the following expression for the Eulerian density:

P (x, t)



|g(x)|=



δ(x −ξ(ξ

, t))P

(ξ

)



|g(ξ

)|d

. (5.26)

The corresponding formula for the Eulerian velocity is contained in the expression

for the current:

P (x, t)



|g(x)|v

(x, t ) =



∂ξ

(ξ

, t)

∂t

δ(x −ξ(ξ

, t))P

(ξ

)



|g(ξ

)|d

. (5.27)

These relations express both the change of picture and the temporal propagation of

the system. Evaluating the integrals, Equations 5.26 and 5.27 are equivalent to the

80 Quantum Trajectories

following local expressions

P (x, t)



|g(x)|=J

−1

(x,t)

(ξ

(x, t ))



|g(ξ

(x, t ))| (5.28)

(x, t ) =

∂ξ

(ξ

, t)

∂t

(x,t)

. (5.29)

These formulas enable us to translate the Lagrangian ﬂow equations into Eulerian

language and solve the latter using the trajectories. Differentiating Equation 5.26 with

respect to t and using Equation 5.27 we deduce the continuity equation

∂P

∂t

√

|g|



∂

∂x



|g|v

) = 0. (5.30)

Next, differentiating Equation 5.27 and using Equations 5.19 and 5.30 we get the

analog of Euler’s classical equation:

∂v

∂t



∂v

∂x



v,σ



vσ

=−



µv

∂

∂x

(Q + V ) +



µv

(5.31)

where Q is given by Equation 5.20 with ξ replaced by x. Finally, the Formula 5.25

for the velocity becomes





µv

∂S(x, t)

∂x

− A

(x)



. (5.32)

Formulas 5.28 and 5.29 give the general solution of the coupled continuity and Euler

Equations 5.30 and 5.31 in terms of the paths and initial wavefunction.

To establish the connection between the Eulerian equations and Schrödinger’s

equation we note that, using Equation 5.32, 5.31 implies

∂S

∂t



µ,v

µv



∂S

∂x

− A



∂S

∂x

− A



+ Q + V = 0 (5.33)

where we have absorbed a function of time in S. Combining Equation 5.33 with Equa-

tion 5.30 (where we substitute Equation 5.32) we ﬁnd that the function Ψ(x, t) =

√

P exp(iS/) obeys the Schrödinger equation in external scalar and vector poten-

tials in general coordinates:

i

∂Ψ

∂t

−

√

|g|



µ,v



∂

∂x

− (i/)A





|g|g

µv



∂Ψ

∂x

− (i/)A



+ V ψ.

(5.34)

We have thus deduced the linear wave equation from the nonlinear collective particle

motion.

The explicit construction of the time-dependent wavefunction in terms of the tra-

jectories follows straightforwardly from Equation 5.28 (which gives the amplitude)

and Equations 5.29 and 5.32. The latter give ∂S/∂x

and S is ﬁxed up to an additive

constant by Equation 5.33.

Quantum Field Dynamics from Trajectories 81

5.4 BOSONS

To make contact with the ﬁeld theory for bosons we let M = R⊗···⊗R with ξ

a set of

Cartesian coordinates and g

µv

= δ

µv

. We assume moreover that the coordinate index

is shorthand for three indices: µ ≡ k

= (k

, k

) where k

=−2πn/L, ...,2πn/L,

i = 1, 2, 3, with L the side of a box of volume L

and N = (2n +1)

. We shall write

k for k

. Choosing A

= 0 and V =



mω

where ω

= k

/2m, the equation

of motion 5.19 of the ξ

th particle becomes

∂

∂t

+ ω

=−

∂Q

∂ξ

, Q =



−

√

∂

√

∂ξ

. (5.35)

The ﬂuid is therefore a continuum of linear oscillators coupled by quantum force terms.

The corresponding classical system is obtained when the quantum force −∂Q/∂ξ

may be neglected.

For notational convenience we write q in place of x in passing to the Eulerian

picture. The Schrödinger Equation 5.34 implied by Equation 5.35 is

i

∂Ψ

∂t





−

∂

∂q

mω



. (5.36)

Writing p

=−i∂/∂q

and using the q

, p

commutation relations, the quantities



|k|q

− p

/i|k|



, a

†



|k|q

+ p

/i|k|



(5.37)

deﬁne a set of bosonic creation and annihilation operators:

, a

†



]

−

= δ



, [a

, a



]

−

=[a

†

, a

†



]

−

= 0. (5.38)

Using these operators, the Hamiltonian in Equation 5.36 becomes

H =



†

(5.39)

where E

= 

/2m. To complete the derivation of the quantum ﬁeld dynamics

we take the limit n →∞and deﬁne a complex ﬁeld operator in space through the

Fourier series

ψ(x) = L

−3/2



ik.x

, k

= 2πn

/L, n

∈ Z, i = 1, 2, 3. (5.40)

The relations Equation 5.38 imply that the ﬁeld operators obey the commutation

relations

[ψ(x), ψ

†



)]

−

= δ(x − x



), [ψ(x), ψ(x



)]

−

=[ψ

†

(x), ψ

†



)]

−

= 0 (5.41)

82 Quantum Trajectories

and the Hamiltonian Equation 5.39 is that of a free boson ﬁeld:

H =

−



†

(x)∇

ψ(x)d

x. (5.42)

We have thus derived the dynamics of a free quantum boson ﬁeld from the trajectory

theory of a continuum of harmonic oscillators coupled by forces derived from the

quantum potential.

5.5 FERMIONS

In the case of fermions we make the following identiﬁcation of the manifold sup-

porting the ﬂow: M = SU(2) ⊗···⊗SU(2). Here the index µ = r, k

where

r = 1, 2, 3 and k

is deﬁned as in Section 5.4. The coordinates ξ

√

3/2|k|

−1

where θ

= (θ

, θ

) are, for each k, a set of three Euler angles (for the deﬁ-

nition see Ref. [6]), and the k-dependent coefﬁcient is chosen in order to achieve a

particular form for the moment of inertia. The metric on M is g

µv

= g



with



= G

(θ

)δ



and





10 0

0 1 cos θ

0 cos θ





, G







10 0

sin

−

cos θ

sin

0 −

cos θ

sin







. (5.43)

Thus g =

det G

(θ

) =

sin

Instead of expressing the law of motion Equation 5.19 directly in terms of the

Euler angles, we can more clearly appreciate the physical model if we use the angular

velocity vector and other quantities carrying vector indices i, j , . . . with respect to

which the metric is δ

. We may pass between the angle and vector descriptions using

the matrices







−cos θ

0 −sin θ

sin θ

0 −sin θ

cos θ

0 −1 −cos θ













−cos θ

cot θ

sin θ

−cosecθ

sin θ

cot θ

cos θ

−cosecθ

cos θ

0 −10







(5.44)

Then, for each k, the angular velocity is ω



∂θ

/∂t and the metrics are given

by G



and δ



. Writing A

(θ

) =−



√

(θ

)

where I

= 3m/2k

we have

Quantum Field Dynamics from Trajectories 83



∂ξ

∂t

=−



i,k

. (5.45)

Next, we introduce a set of angular momentum operators for each k,

=−i



∂/∂θ

, i = 1, 2, 3, (5.46)

so that

−



µ,v

µv

(ξ)

∂P

∂ξ

∂P

∂ξ

=−



r,s,k





(θ

)

∂P

∂θ

∂P

∂θ



i,k





P . (5.47)

Putting all this together in Equation 5.15, the Lagrangian may be written

L =



(θ

)



i,k



P −I

(θ

)ω

− V (θ

)



dΩ

(5.48)

where dΩ

= sin θ

dθ

. This is the Lagrangian of a collection of spher-

ical rotators of moment of inertia I

subject to external magnetic-type ﬁelds B

and

a scalar potential V . Their quantum-mechanical character is expressed through the

quadratic interaction potential and the constraints implied by the initial wavefunc-

tion. In terms of these variables the (ﬁnal) Lorentz force-type term in Equation 5.19

contributes several terms including a Larmor-type torque and the equation of motion

of the θ

th particle is

∂ω

∂t



j,l

ijl

iI

(Q + V ) +

i



(

−

). (5.49)

The ﬂuid is therefore a continuum of spin

rotators subject to Larmor (the ﬁrst

term on the right-hand side of Equation 5.49), quantum and external torques. The

corresponding classical system is obtained when the quantum torque

Q/i may

be neglected.

The justiﬁcation for this construction is seen on passing to the Eulerian picture

where we write x

√

3/2|k|

−1

. First, we note that the Schrödinger Equa-

tion 5.34 implied by Equation 5.49 describes a set of spherical rotators, in external

“magnetic” and scalar potentials, in the angular coordinate representation:

i

∂Ψ

∂t



i,k

(

+ I

)(

+ I

)Ψ + V Ψ (5.50)