Chattaraj P.K. (Ed.) Quantum Trajectories

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14 Quantum Trajectories

spin-statistics rule asserts that every species with integer spin (0, 1, 2, ...)isbosonic

and every species with half-odd spin (

, ...)isfermionic.

In Bohmian mechanics for identical particles, one uses the same type of wave

function as in ordinary quantum mechanics, and the same equation of motion (and

Schrödinger equation) as in Bohmian mechanics for distinguishable particles. That is,

we include the symmetrization postulate among the postulates of Bohmian mechanics.

It is a traditional claim in textbooks on quantum mechanics that the lack of precise

trajectories in orthodox quantum mechanics is the reason for the symmetrization pos-

tulate. If the particles had trajectories, it is suggested, then they would automatically

be distinguishable. From Bohmian mechanics with the symmetrization postulate we

see that this suggestion is incorrect.

On the contrary, the Bohmian trajectories actually enhance our understanding of

the symmetrization postulate. We start from the following observation:

If(...Q

(0)...Q

(0)...)evolves to ( . . . Q

(t)...Q

(t)...)

then ( . . . Q

(0)...Q

(0)...)evolves to ( . . . Q

(t)...Q

(t)...). (1.41)

In other words, it is unnecessary to specify the labelling of the particles. That seems

very appropriate as the labelling is unphysical. The fact expressed by Equation 1.41

follows from the symmetry of the velocity vector ﬁeld,

(...q

...q

...)= v

(...q

...q

...), (1.42)

(...q

...q

...)= v

(...q

...q

...) fori = k = j , (1.43)

which can easily be checked from the formula 1.4 for v

together with Equations 1.39

or 1.40. For another way of putting the fact expressed by Equation 1.41, let us consider

a system of N identical particles. The natural conﬁguration space is

={Q ⊂ R

: #Q = N }, (1.44)

the set of all N -element subsets of R

. While an element of R

is an ordered

conﬁguration (Q

, ...,Q

), an element of

is an unordered conﬁguration

, ...,Q

}. Since the labels need not be speciﬁed, any point Q(0) ∈

initial condition will uniquely deﬁne a curve t → Q(t) ∈

. So for symmetric

or anti-symmetric wave functions, Bohmian mechanics works on the natural conﬁg-

uration space of identical particles. This fact can be regarded as something like an

explanation, or derivation, of the symmetrization postulate. For a deeper discussion

see [7].

1.14 FURTHER READING

Concerning the extension of Bohmian mechanics to quantum ﬁeld theory, see [6, 10].

Concerning the extension of Bohmian mechanics to relativistic space-time, see the

review article [11] and references therein. Concerning quantum non-locality, see

Bell’s book [3], which we highly recommend also about foundations of quantum

mechanics in general.

Bohmian Trajectories as the Foundation of Quantum Mechanics 15

BIBLIOGRAPHY

1. V. Allori, D. Dürr, S. Goldstein, and N. Zanghì: Seven Steps Towards the Classical World.

Journal of Optics B 4: 482–488 (2002). http://arxiv.org/abs/quant-ph/0112005.

2. J. S. Bell: On the Problem of Hidden Variables in Quantum Mechanics. Reviews of Modern

Physics 38: 447–452 (1966). Reprinted as chapter 1 of [3].

3. J. S. Bell: Speakable and Unspeakable in Quantum Mechanics. Cambridge: University

Press (1987).

4. D. Bohm: A Suggested Interpretation of the Quantum Theory in Terms of “Hidden” Vari-

ables, I and II. Physical Review 85: 166–193 (1952).

5. L. de Broglie: in Electrons et Photons: Rapports et Discussions du Cinquième Conseil

de Physique tenu à Bruxelles du 24 au 29 Octobre 1927 sous les Auspices de l’Institut

International de Physique Solvay, Paris: Gauthier-Villars (1928). English translation in

G. Bacciagaluppi and A. Valentini: Quantum Theory at the Crossroads: Reconsidering

the 1927 Solvay Conference, Cambridge: University Press (2009). http://arxiv.org/abs/

quant-ph/0609184.

6. D. Dürr, S. Goldstein, R. Tumulka, and N. Zanghì: Bohmian Mechanics and Quan-

tum Field Theory. Physical Review Letters 93: 090402 (2004). http://arxiv.org/abs/

quant-ph/0303156.

7. D. Dürr, S. Goldstein, J. Taylor, R. Tumulka, and N. Zanghì: Quantum Mechanics

in Multiply-Connected Spaces. Journal of Physics A: Mathematical and Theoretical

40: 2997–3031 (2007). http://arxiv.org/abs/quant-ph/0506173.

8. D. Dürr, S. Goldstein, and N. Zanghì: Quantum Equilibrium and the Origin of Abso-

lute Uncertainty. Journal of Statistical Physics 67: 843–907 (1992). http://arxiv.org/abs/

quant-ph/0308039.

9. C. L. Lopreore and R. E. Wyatt: Quantum Wave Packet Dynamics with Trajectories.

Physical Review Letters 82: 5190–5193 (1999).

10. W. Struyve: Field beables for quantum ﬁeld theory. Preprint (2009). http://arxiv.org/

abs/0707.3685.

11. R. Tumulka: The “Unromantic Pictures” of Quantum Theory. Journal of Physics A: Math-

ematical and Theoretical 40: 3245–3273 (2007). http://arxiv.org/abs/quant-ph/0607124.

The Equivalence Postulate

of Quantum Mechanics:

Main Theorems

Alon E. Faraggi and Marco Matone

CONTENTS

2.1 The Hamilton–Jacobi Equation and Coordinate Transformations..................17

2.2 The Equivalence Postulate..............................................................................19

2.3 Proof of Theorem 1.........................................................................................24

2.4 Proof of Theorem 2.........................................................................................30

Bibliography ............................................................................................................37

2.1 THE HAMILTON–JACOBI EQUATION AND COORDINATE

TRANSFORMATIONS

The Hamilton–Jacobi (HJ) equation for a one-dimensional system is obtained by

considering the canonical transformation (q, p) → (Q, P ) so that the old Hamiltonian

H maps to a trivialized one, that is

H = 0. The old and new momenta are expressed

in terms of the generating function of such a transformation, the Hamilton’s principal

function p =

∂S

∂q

, P = cnst =−

∂S

∂Q

Q=cnst

that satisﬁes the classical HJ equation



q, p =

∂S

∂q

, t



∂S

∂t

= 0.

In the case of a time-independent potential the time-dependence in the Hamilton’s

principal function S

is linear, that is S

(q, Q, t) = S

(q, Q) − Et, with E the

energy of the stationary state. It follows that S

, called Hamilton’s characteristic

function, or reduced action, satisﬁes the classical stationary HJ equation (CSHJE)



q, p =

∂S

∂q



− E = 0,

that is (W(q) ≡ V (q) −E)



∂S

∂q



+ W = 0.

18 Quantum Trajectories

Note that the canonical transformation (q, p) → (Q, P ) treats p and q as indepen-

dent variables. Following [1–6] we now formulate a similar question to that leading

to the CSHJE, but considering the transformation on q, with the one on p induced by

the relation

p =

∂S

∂q

More precisely, given a one-dimensional system, with time-independent potential

(the higher dimensional time-dependent case is considered in [7]) we look for the

coordinate transformation q → q

such that

(q)

Coord. Transf.

←→

cl 0

), (2.1)

with

cl 0

) denoting the reduced action of the system with vanishing Hamilto-

nian. Note that in Equation 2.1 we required that this transformation be an invertible

one. This is an important point since by compositions of the maps it follows that if

for each system there is a coordinate transformation leading to the trivial state, then

even two arbitrary systems are equivalent under coordinate transformations. Impos-

ing this apparently harmless analogy immediately leads to rather peculiar properties

of classical mechanics (CM). First, it is clear that such an equivalence principle

cannot be satisﬁed in CM, in other words given two arbitrary systems a and b, the

condition

cl b

) = S

cl a

), (2.2)

cannot be generally satisﬁed. In particular, since

cl 0

) = cnst,

it is clear that Equation 2.1 is a degenerate transformation. However, in principle, by

itself the failure of Equation 2.2 for arbitrary systems would be a possible natural

property. Nevertheless, a more careful analysis shows that such a failure is strictly

dependent on the choice of the reference frame. This is immediately seen by consid-

ering two free particles of mass m

and m

moving with relative velocity v. For an

observer at rest with respect to particle a the two reduced actions are

cl a

) = cnst, S

cl b

) = m

It is clear that there is no way to have an equivalence under coordinate transformations

by setting S

cl b

) = S

cl a

). This means that at the level of the reduced action there

is no coordinate transformation making the two systems equivalent. However, note

that this coordinate transformation exists if we consider the same problem described

by an observer in a frame in which both particles have a non-vanishing velocity so that

the two particles are described by non-constant reduced actions. Therefore, in CM,

it is possible to connect different systems by a coordinate transformation except in

the case in which one of the systems is described by a constant reduced action. This

means that in CM equivalence under coordinate transformations is frame-dependent.

In particular, in the CSHJE description there is a distinguished frame. This seems

The Equivalence Postulate of Quantum Mechanics 19

peculiar as on general grounds what is equivalent under coordinate transformations

in all frames should remain so even in the one at rest

2.2 THE EQUIVALENCE POSTULATE

The above investigation already suggests that the concept of point particle itself cannot

be consistent with the equivalence under coordinate transformations. In particular, it

suggests that the system where a particle is at rest does not exist at all. If this were

the case, then the above critical situation would not occur simply because the reduced

action is never a constant. This should result in two main features. First the classical

concept of point particle should be reconsidered, and secondly the CSHJE should be

modiﬁed accordingly. A natural suggestion would be to consider particles as a kind of

string with a lower bound on the vibrating modes in such a way that there is no way

to deﬁne a system where the particle is at rest. It should be observed that this kind of

string may differ from the standard one, rather its nature may be related to the fact

that in general relativity it is impossible to deﬁne the concept of relative stability of

a system of particles.

In Ref. [8–10] it was suggested that quantum mechanics and gravity are intimately

related. In particular, it was argued that the quantum HJ equation of two free particles,

which is attractive, may generate the gravitational potential. This is a consequence

of the fact that the quantum potential is always non-trivial even in the case of the

free particle. It plays the role of intrinsic energy and may in fact be at the origin of

fundamental interactions.

The uniﬁcation of quantum mechanics and general relativity is the central question

of theoretical physics. This problem hinges on the viability of the prevailing theories

of matter and interactions at the micro-scale, and of the cosmos at the macro-scale. The

Galilean paradigm of modern science drives the search for a mathematical formulation

of the synthesis of quantum gravity. In such a context string theory provides an attempt

at a self-consistent mathematical formulation of quantum gravity.

String theory provides a perturbatively ﬁnite S-matrix approach to the calcula-

tion of string scattering amplitudes. Due to its unique world-sheet properties, string

theory admits a discrete particle spectrum. It accommodates the gauge bosons and

fermion matter states that form the bedrock of modern particle physics, as well as

a massless spin 2 symmetric state, which is interpreted as the gravitational force

mediation ﬁeld. Consequently, string theory enables the construction of models that

admit the structures of the standard particle model and enables the development of a

phenomenological approach to quantum gravity. The state of the art in this regard is

the construction of minimal heterotic string standard models, which produces in the

observable standard model charged sector solely the spectrum of the minimal super-

symmetric standard model [11–13]. Progress in the understanding of string theory was

obtained by the observation that the ﬁve 10-dimensional string theories, as well as

11-dimensional supergravity, can be connected by perturbative and non-perturbative

duality tranformations. However, this observation does not provide a rigorous formu-

lation of quantum gravity, akin to the formulations of general relativity and quantum

20 Quantum Trajectories

mechanics, which follow from the equivalence principle in the former and the prob-

ability interpretation of the wavefunction in the later.

Let us start imposing the equivalence under coordinate transformations. The key

point is to consider, like in general relativity, the (analog of the) reduced action as a

scalar ﬁeld under coordinate transformations.

We postulate that for any pair of one-particle states there exists a ﬁeld S

such that

) = S

) (2.3)

is well deﬁned. We also require that, in a suitable limit, S

reduces to S

. Equa-

tion 2.3 can be considered as the scalar hypothesis. Since the conjugate momentum

is deﬁned by

∂

∂q

(q),

it follows by Equation 2.3 that the conjugate momenta p

and p

are related by a

coordinate transformation

= Λ

, (2.4)

where Λ

= ∂q

/∂q

. Note that we have the invariant

= p

. (2.5)

Since Equation 2.3 holds for any pair of one-particle states, we have Det

Λ(q) = 0, ∀q.

The scalar hypothesis Equation 2.3 implies that two one-particle states are always

connected by a coordinate transformation; for this reason we may equivalently con-

sider Equation 2.3 as imposing an equivalence postulate (EP). In particular, while in

arbitrary dimension the coordinate transformation is given by imposing Equation 2.4,

in the one-dimensional case the scalar hypothesis implies

= S

−1

◦ S

We now consider the consequences of the EP Equation 2.3. Let us denote by H

the space of all possible W ≡ V − E. We also call v-transformations those leading

from one system to another. Equation 2.3 is equivalent to requiring that

For each pair W

, W

∈ H, there is a v-transformation such that

(q) −→ W

) = W

). (2.6)

This implies that there always exists the trivializing coordinate q

for which

W(q) −→ W

), where

) ≡ 0.

In particular, since the inverse transformation should exist as well, it is clear that the

trivializing transformation should be locally invertible. We will also see that since

classically W

is a ﬁxed point, implementation of Equation 2.6 requires that W states

transform inhomogeneously.

The Equivalence Postulate of Quantum Mechanics 21

The fact that the EP cannot be consistently implemented in CM is true in any

dimension. To show this, let us consider the coordinate transformation induced by

the identiﬁcation

cl v

) = S

(q) . (2.7)

Then note that the CSHJE



k=1

(∂

(q))

+ W(q) = 0, (2.8)

provides a correspondence between W and S

that we can use to ﬁx, by consis-

tency, the transformation properties of W induced by that of S

. In particular, since

cl v

) must satisfy the CSHJE



k=1

(∂

cl v

))

+ W

) = 0, (2.9)

by Equation 2.7 we have

∂S

cl v

)

∂q

= Λ

∂S

(q)

∂q

. (2.10)

Let us set (p

|p) =p

Λp/p

p. By Equations 2.8 through 2.10, we have

W(q) −→ W

) = (p

|p)W(q), so that

) −→ W

) = (p

) = 0.

Thus we have [1–6]:

W states transform as quadratic differentials under classical v-maps. It follows that

is a ﬁxed point in H. Equivalently, in CM the space H cannot be reduced to a point

upon factorization by the classical v-transformations. Hence, the EP Equation 2.6

cannot be consistently implemented in CM. This can be seen as the impossibility

of implementing covariance of CM under the coordinate transformation deﬁned by

Equation 2.7.

It is therefore clear that in order to implement the EPwe have to deform the CSHJE.

As we will see, this requirement will determine the equation for S

In Refs [1–6] the function T

(p), deﬁned as the Legendre transform of the reduced

action, was introduced:

(p) = q

− S

(q), S

(q) = p

− T

(p).

While S

(q) is the momentum generating function, its Legendre dual T

(p)isthe

coordinate generating function

∂S

∂q

, q

∂T

∂p

22 Quantum Trajectories

Note that adding a constant to S

does not change the dynamics. Then, the most

general differential equation S

should satisfy has the structure

F(∇S

, ∆S

, ...)= 0. (2.11)

Let us write down Equation 2.11 in the general form



k=1

(∂

(q))

+ W(q) +Q(q) = 0.

The transformation properties of W + Q under the v-maps are determined by the

transformed equation



k=1

(∂

))

/2m + W

) + Q

) = 0, (2.12)

so that

) + Q

) = (p

|p)

[

W(q) +Q(q)

]

. (2.13)

A basic guidance in deriving the differential equation for S

is that in some limit

it should reduce to the CSHJE. In Refs [1–10] it was shown that the parameter

which selects the classical phase is the Planck constant. Therefore, in determining the

structure of the Q term we have to take into account that in the classical limit

lim

→0

Q = 0. (2.14)

The only possibility to reach any other state W

= 0 starting from W

is that it

transforms with an inhomogeneous term. Namely as W

−→ W

) = 0, it follows

that for an arbitrary W

state

) = (p

) + (q

), (2.15)

and by Equation 2.13

) = (p

) − (q

). (2.16)

Let us stress that the purely quantum origin of the inhomogeneous term (q

)is

particularly transparent once one consider the compatibility between the classical

limit (Equation 2.14) and the transformation properties of Q in Equation 2.16.

The W

state plays a special role. Actually, setting W

= W

in Equation 2.15

yields

) = (q

so that, according to the EP Equation 2.6, all the states correspond to the inhomoge-

neous part in the transformation of the W

state induced by some v-map.

Let us denote by a, b, c, . . . different v-transformations. Comparing

) = (p

) + (q

) = (q

), (2.17)

The Equivalence Postulate of Quantum Mechanics 23

with the same formula with q

and q

interchanged we have

) =−(p

)(q

), (2.18)

in particular (q;q) = 0. More generally, imposing the commutative diagram of maps



−→



that is comparing

) = (p

) + (q

)

= (p

) + (p

)(q

) + (q

with Equation 2.17, we obtain the basic cocycle condition

) = (p

)

[

) + (q

)

]

, (2.19)

which expresses the essence of the EP. In the one-dimensional case we have

) =



∂



) + (q

). (2.20)

It is well known that this is satisﬁed by the Schwarzian derivative. However, it turns

out that it is essentially the unique solution. More precisely [1–6],

Theorem 2.1

Equation 2.20 deﬁnes the Schwarzian derivative up to a multiplicative constant and

a coboundary term.

Since the differential equation for S

should depend only on ∂

, k ≥ 1, it follows

that the coboundary term must be zero, so that [1–6]

) =−

, q

where {f (q), q}=f





− 3(f





)

/2 is the Schwarzian derivative and β is a

non-vanishing constant that we identify with . As a consequence, S

satisﬁes the

quantum stationary Hamilton–Jacobi equation (QSHJE) [1–6]



∂S

(q)

∂q



+ V (q) −E +



, q}=0. (2.21)

Note that ψ = S



−1/2

(Ae

−



+ Be



) solves the Schrödinger equation (SE)



−



∂

∂q

+ V



ψ = Eψ. (2.22)