Hamiltonian Formulation of the Pilot-Wave Theory Has Also Been Developed by Holland [9]

Home , Pilot wave theory

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In the standard approach to quantum mechanics the wave function provides a complete description of any system and is used to calculate probability distributions for observ- ables associated with the system. In the pilot-wave theory pioneered by de Broglie [1, 2] in the 1920’s, particles have definite positions and velocities and are “guided” by the wave function, which satisfies Schrodinger’s equation. A similar approach was developed by Bohm [3, 4] in the early 1950’s (see [5] and [6] for extensive reviews). For a single particle in the non-relativistic theory the velocity of the particle is given by dX~ = J~(X,~ t) , (1) dt where X~ is the particle’s position and h¯ J~ = ψ∗ ~ ψ ψ ~ ψ∗ . (2) 2im ψ 2 ∇ − ∇ | | h i Particle trajectories are, therefore, integral curves to the vector field J~. In this paper I first consider the pilot-wave theory for a non-relativistic particle in a potential. I construct a Hamiltonian that gives Schrodinger’s equation and the guidance equation for the particle. This involves imposing a constraint and the use of Dirac’s method of dealing with constrained dynamical systems [7, 8]. I then find the Hamiltonian for a relativistic particle in Dirac’s theory and for a quantum scalar field. A Hamiltonian formulation of the pilot-wave theory has also been developed by Holland [9]. His approach is significantly different from the approach taken in this paper and will be discussed at the end of section 2.

2 Non-Relativistic Pilot-Wave Theory

In this section I construct a Hamiltonian that gives Schrodinger’s equation and the guidance equation for a single particle and for a collection of particles. The Lagrangian

1 ∂ψ ∂ψ∗ h¯2 L = ih¯ ψ∗ ψ ~ ψ ~ ψ∗ V ψ∗ψ (3) 2 " ∂t − ∂t # − 2m∇ · ∇ −

gives Schrodinger’s equation

2 ∂ψ h¯ ih¯ = 2ψ + V ψ (4) ∂t −2m∇ under a variation with respect to ψ∗ and its complex conjugate under a variation with respect to ψ.

2 ∗ The canonical momenta Πψ = ∂L/∂ψ˙ and Πψ∗ = ∂L/∂ψ˙ are given by

1 ∗ 1 Π = ihψ¯ and Π ∗ = ihψ¯ . (5) ψ 2 ψ −2 We therefore have the primary constraints

1 ∗ 1 φ1 = Π ihψ¯ 0 and φ2 = Π ∗ + ihψ¯ 0 , (6) ψ − 2 ≈ ψ 2 ≈ where denotes a weak equality, which can only be imposed after the Poisson brackets have been≈ evaluated. These constraints satisfy

φ1(~x),φ1(~y) = φ2(~x),φ2(~y) =0 (7) { } { } and 3 φ1(~x),φ2(~y) = ihδ¯ (~x ~y) . (8) { } − − These constraints are, therefore, second class constraints. ∗ The canonical Hamiltonian density h = Π ψ˙ + Π ∗ ψ˙ L is given by C ψ ψ − 2 h¯ ∗ ∗ ∗ h = ~ ψ ~ ψ + V ψ ψ + φ1ψ˙ + φ2ψ˙ (9) C 2m∇ · ∇ and the total Hamiltonian density is given by

2 h¯ ∗ ∗ h = ~ ψ ~ ψ + V ψ ψ + u1φ1 + u2φ2 , (10) T 2m∇ · ∇

where u1 and u2 are undetermined parameters. For consistency we require that

φ˙1 = φ1,H 0 and φ˙2 = φ2,H 0 , (11) { T } ≈ { T } ≈ 3 where HT = hT d x is the total Hamiltonian. These two equations give

R 2 i h¯ 2 u1 = ψ V ψ (12) h¯ "2m∇ − # and 2 i h¯ 2 ∗ ∗ u2 = ψ V ψ . (13) −h¯ "2m∇ − #

Substituting these expressions for u1 and u2 into HT and integrating by parts gives

2 2 i h¯ 2 h¯ 2 ∗ ∗ 3 HT = Πψ ψ V ψ Πψ∗ ψ V ψ d x . (14) h¯ Z " 2m∇ − ! − 2m∇ − !#

3 The equation of motion for ψ is 2 i h¯ 2 ψ˙ = ψ,HT = ψ V ψ , (15) { } h¯ 2m∇ − !

which is Schrodinger’s equation. The equation of motion for Πψ is 2 i h¯ 2 Π˙ ψ = Πψ,HT = Πψ V Πψ . (16) { } h¯ 2m∇ − !

From (6) we see that this is Schrodinger’s equation for ψ∗. Similar results hold for ψ˙ ∗ and for Π˙ ψ∗ . The constraints φ1 and φ2 can be promoted to strong equations if the Poisson bracket is replaced by the Dirac bracket. ih¯ ~ ~ i A similar Hamiltonian density h = 2m Π ψ h¯ V Πψ appears in Schiff [10], but he does not follow Dirac’s procedure to− obtain∇ h· ∇. Gergely− [11] follows Dirac’s approach 2 h¯ ∗ ∗ ∗ and finds the Hamiltonian density h = ~ ψ ~ ψ +V ψ ψ +ψφ˙ 1 +ψ˙ φ2. Schrodinger’s 2m ∇ ·∇ equation and its complex conjugate then follow from the conditions φ˙1 0 and φ˙2 0 ≃ ≃ (see [12] for a discussion on the Hamiltonian formulation of the DKP equation, which is also a first order equation, using Dirac’s procedure). In the pilot-wave theory the wave function is written as

ψ = ReiS/h¯ , (17) where R and S are real functions. The velocity of the particle is taken to be [2, 3]

dX~ 1 = ~ S(X,~ t) , (18) dt m∇ where X~ (t) is the particle’s position. The velocity can also be written as

dX~ ~j h¯ = = ψ∗ ~ ψ ψ ~ ψ∗ , (19) dt ψ∗ψ 2im ψ 2 ∇ − ∇ | | h i where ~j is the probability current density. This equation of motion follows from the Hamiltonian ~p H = ~ S(X,~ t) . (20) p m · ∇ To see this consider the equations of motion

˙ k k 1 ∂S X = X ,Hp = k (21) { } m "∂x #~x=X~

and 2 pl ∂ S p˙k = pk,Hp = k l . (22) { } −m "∂x ∂x #~x=X~

4 Equation (21) is the correct equation of motion for the particle. Equation (22) gives the equation of motion for pk, but pk is not related to the particle velocity in this theory. We can therefore solve for the particle’s trajectory using (21) alone and ignore (22). Now consider the Hamiltonian for the ﬁeld and particle

H = HT + HP . (23)

The equations of motion ψ˙ = ψ,H , X˙ k = Xk,H andp ˙ = p ,H are the same as { } { } k { k } above. However the equation of motion for Πψ is altered by the addition of Hp to HT : p Π˙ = Π ,H = Π ,H + k Π , ∂ S . (24) ψ { ψ } { ψ T } m { ψ k } To evaluate Π , kS it is convenient to write ∂ S as { ψ ∇ } k ∗ ~ h¯ ∂kψ(~x, t) ∂kψ (~x, t) 3 ~ 3 ∂kS(X, t)= ∗ δ (~x X)d x . (25) 2i Z " ψ(~x, t) − ψ (~x, t) # − The last term in (24) given by p p ∂ k Π (~x), ∂ S(X,~ t) = k δ3(~x X~ ). (26) m { ψ k } mψ(~x, t) ∂xk −

This term can be eliminated by imposing the constraint pk 0. For consistency we require thatp ˙ = p ,H 0. From equation (22) we see≈ that this is satisﬁed, so k { k } ≈ that no new constraints arise. The Hamiltonian H plus the constraint pk 0 therefore generates Schrodinger’s equation and the equation of motion for the particle.≈ The generalization of (23) to a system of N particles is straightforward. The Hamil- tonian is given by

N 2 N 2 N i h¯ 2 h¯ 2 ∗ ∗ 3 3 ~pk ∗ ~ H = Πψ kψ V ψ Πψ kψ V ψ d x1...d xN + kS, h¯ Z " k=1 2mk ∇ − ! − k=1 2mk ∇ − !# k=1 mk ·∇ X X X(27) where ~ S can also be written as ∇k N ~ h¯ ∗ ~ ~ ∗ kS = 2 ψ kψ ψ kψ , (28) ∇ =1 2i ψ ∇ − ∇ kX | | h i

ψ = ψ(~x1..., ~xN ) and V = V (~x1..., ~xN , t). This Hamiltonian plus the constraints ~pk 0 gives both the ﬁeld and particle equations of motion. ≈ A Hamiltonian formulation of the pilot-wave theory has also been developed by ¯ Holland [9]. In his approach the wave function is written as ψ = √ρeiS/h and the Hamiltonian is given by

1 ∂ ∂S 1 ∂S ∂S 3 ′ Htot = H + πρ ′ ρ ′ πS ′ ′ + Q + V d q , (29) Z (− " m ∂qi ∂qi !# − "2m ∂qi ! ∂qi ! #)

5 where p2 H = i + V + Q, (30) 2m Xi th pi is the momentum of the i particle, V is the classical potential and Q is the quantum potential which is given by 2 2 h¯ ∂ √ρ Q = 2 . (31) −2m√ρ ∂qi Hamilton’s equations for S˙ andρ ˙ give Schrodinger’s equation and Hamilton’s equations for the particle degrees of freedom are given by ∂ q˙i = pi/m and p˙i = [V + Q] . (32) −∂qi Combining these two equations gives ∂ mqï = [V + Q] , (33) −∂qi which is the second order form of the theory. This equation can be derived from the first order guidance equation, but it is not equivalent to the guidance equation. Holland then considers Liouville’s equation and shows, for pure states, that the momentum is constrained to satisfy the guidance equation. This approach differs significantly from the approach taken in this paper.

3 Relativistic Pilot-Wave Theory

In this section I construct a Hamiltonian that gives the Dirac equation plus the guidance equation. The Lagrangian for the Dirac ﬁeld coupled to the electromagnetic ﬁeld is given by

i ∗ L = ψγ¯ µD ψ (D ψ¯)γµψ mψψ¯ , (34) 2 µ − µ − h i where Dµ = ∂µ + ieAµ and I have takenh ¯ = c = 1. There are two primary constraints

1 0 1 0 φ1 = Π iψγ¯ 0 and φ2 = Π ¯ + iγ ψ 0 (35) ψ − 2 ≈ ψ 2 ≈ that satisfy φ1 (~x),φ1 (~y) = φ2 (~x),φ2 (~y) =0 (36) { k ℓ } { k ℓ } and 0 3 φ1 (~x),φ2 (~y) = i(γ ) δ (~x ~y) . (37) { k ℓ } − ℓk −

6 The constraints are, therefore, second class. Following the same procedure that was used for Schrodinger’s equation gives

0 0 3 H = i Π γ iγk∂ ψ eA γµψ mψ + i(∂ ψ¯)γk + eψA¯ γµ + mψ¯ γ Π ¯ d x. T ψ k − µ − k µ ψ Z n h i h i o (38) In the pilot-wave theory the velocity of the particle is given by [13, 14]

dXµ = J µ (39) dτ where ¯ µ µ ψγ ψ µ J = , J Jµ =1 , (40) √a2 + b2 a = ψψ¯ and b = iψγ¯ 5ψ . (41) The particle Hamiltonian is µ HP = pµJ (42) and the Hamiltonian for the ﬁeld and particle is given by

H = HT + Hp . (43)

As in the non-relativistic approach we must impose p 0 so that addition of H to H µ ≈ p T does not alter the ﬁeld equations for Πψ and Πψ¯.

4 Pilot-Wave Approach in Scalar Quantum Field Theory

In this section I consider a massive scalar field and find a Hamiltonian that gives Schrodinger’s equation plus the guidance equation for the field (see [15, 16] for discussions on the pilot-wave theory for scalar fields). Consider a real scalar field φ(xµ) with the Lagrangian 1 1 L = ∂ φ∂µφ m2φ2 . (44) 2 µ − 2 The canonical momentum is given by ∂φ Π= (45) ∂t and the Hamiltonian is given by 1 H = Π2 +( φ)2 + m2φ2 d3x . (46) 2 ∇ Z h i 7 Schrodinger’s equation for this theory is given by

2 ∂Ψ 1 δ 2 2 2 3 i = 2 +( φ) + m φ d x Ψ . (47) ∂t 2 Z "−δφ ∇ # The wave function can be written as

Ψ= ReiS (48) and the equation of motion is taken to be ∂φ δS Π= = . (49) ∂t δφ Consider confining the field to a box of length L with periodic boundary conditions. The field can be written as

1 ik·x φ(~x, t)= qk(t)e , (50) sV k X where 2π k = (n , n , n ) , (51) L x y z ∗ ∗ nx, ny and nz integers, qk = q−k and pk = p−k. Schrodinger’s equation is given by

2 ∂Ψ ∂ 2 2 ∗ i = ∗ +(k + m )qkqk Ψ , (52) ∂t k 2 "−∂qk∂qk # X/ ∗ where Ψ = Ψ(qk, qk, t) and k/2 indicates that the sum is carried out over half of the values of k. A Lagrangian that produces (52) is

∗ ∗ 1 ∗ ∂Ψ ∂Ψ ∂Ψ ∂Ψ 2 2 ∗ ∗ L = i Ψ Ψ ∗ +(k + m )qkqkΨΨ . (53) 2 " ∂t − ∂t # − k 2 "∂qk ∂qk # X/ Following the same procedure that was discussed in section 2 leads to the constraints

1 ∗ 1 ΠΨ ih¯Ψ ΠΨ∗ ih¯Ψ (54) ≈ 2 ≈ −2 and to the total Hamiltonian 2 2 ∗ ∂ Ψ 2 2 ∗ ∂ Ψ 2 2 ∗ ∗ ∗ HT = i ΠΨ ∗ (k + m )qkqkΨ ΠΨ∗ ∗ (k + m )qkqkΨ dqdq , k 2 ( "∂qk∂qk − # − " ∂qk∂qk − #) X/ Z (55)

8 ∗ ∗ where dq = Πk/2dqk and dq = Πk/2dqk. This Hamiltonian gives the correct equations of motion assuming Gauss’ theorem (and the vanishing of surface terms) in inﬁnite dimensional spaces. The guidance equations are given by

∂S ∗ ∂S q˙k = ∗ and q˙k = . (56) ∂qk ∂qk This equation of motion follows from the Hamiltonian

∂S ∗ ∂S Hp = pk ∗ + pk , (57) k 2 " ∂qk ∂qk # X/ where ∗ ∗ qk,pl = qk,pl = δkl (58) { } { } and all other Poisson brackets vanish. The equations of motion of the particle and ﬁeld follow from the Hamiltonian H = HT + Hp (59) ∗ plus the constraints pk 0 and pk 0. ≈ ≈ 5 Conclusion

In this paper I showed that the Hamiltonian

N 2 N 2 N i h¯ 2 h¯ 2 ∗ ∗ 3 3 ∗ ~ H = Πψ kψ V ψ Πψ kψ V ψ d x1...d xN + ~pk kS h¯ Z " k=1 2mk ∇ − ! − k=1 2mk ∇ − !# k=1 ·∇ X X X(60) plus the constraints ~pk 0 gives Schrodinger’s equation for a collection of non-relativistic particle in a potential and≈ the guidance equation

dX~ k = ~ S(X,~ t) , (61) dt ∇k

th where X~ k is the position of the k particle. It is important to note that the canonical momenta ~p are not related to ~v and it is consistent to impose the constraints ~p 0. k k k ≈ I then showed that

0 0 3 H = i Π γ iγk∂ ψ eA γµψ mψ + i(∂ ψ¯)γk + eψA¯ γµ + mψ¯ γ Π ¯ d x+p J µ, ψ k − µ − k µ ψ µ Z n h i h i o (62) where ψγ¯ µψ J µ = , (63) √a2 + b2 a = ψψ¯ and b = iψγ¯ 5ψ (64)

9 plus the constraint p 0 gives the Dirac equation and the guidance equation µ ≈ dXµ = J µ . (65) dτ Lastly I considered the massive Klein-Gordon equation as a quantum ﬁeld (not as a single particle wave equation). After decomposing the ﬁeld into normal modes

1 ik·x φ(~x, t)= qk(t)e (66) sV k X I showed that the Hamiltonian

∂2Ψ 2 2 ∗ ∂2Ψ∗ 2 2 ∗ ∗ ∗ H = i k 2 ΠΨ ∗ (k + m )qkqkΨ ΠΨ∗ ∗ (k + m )qkqkΨ dqdq / ∂qk∂qk ∂qk∂qk − − − P R ∂S ∗ ∂S + k 2 pk ∗ + pk , / ∂qk ∂qk P (67) ∗ ∗ ∗ where dq = Πk/2dqk and dq = Πk/2dqk, plus the constraints pk 0 and pk 0 gives Schrodinger’s equation and the guidance equations ≈ ≈

∂S ∗ ∂S q˙k = ∗ and q˙k = . (68) ∂qk ∂qk Acknowledgements

This research was supported by the Natural Sciences and Engineering Research Council of Canada.

References

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[16] P. Holland, Phys. Rept. 224, 95 (1993)