QUANTUMMECHANICAL BEHAVIOUR IN A DETERMINISTIC MODEL. G. t Hooft

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1 QUANTUMMECHANICAL BEHAVIOUR IN A DETERMINISTIC MODEL G. t Hooft Institute for Theoretical Physics University of Utrecht, P.O.Box TA Utrecht, the Netherlans g.thooft@fys.ruu.nl THU-96/39 quant-ph/ Abstract A eterministic moel with a large number of continuous an iscrete egrees of freeom is escribe, an a statistical treatment is propose. The moel exactly obeys a Schröinger equation, which has to be interprete exactly accoring to the Copenhagen prescriptions. After applying a Hartree-Fock approximation, the moel appears to escribe genuine quantum particles that coul be use as a starting point for fiel variables in a quantum fiel theory. In the eterministic moel it is essential that information loss occurs, but the corresponing quantum system is unitary an exactly preserves information. This paper is intene to exhibit the mathematical nature of a moel that coul be use for further stuy of the quantum mechanical nature of this worl, but the philosophical implications will not be aresse, nor will we iscuss possible implications in connection with the usual quantummechanics paraoxes such as the Einstein Rosen Poolski paraox these are postpone to a more elaborate publication. In the present paper, the reaer is invite to raw his or her own conclusions concerning the relevance of this moel. First, consier one continuous parameter q(t) R (bounary conitions on q will be postpone to later) an one iscrete parameter s = ±1, an let them obey the (eterministic) time evolution equations q(t) =s; t s=0. (1) t 1

2 This means that we have a particle moving along the real line with velocity either +1 or 1. Defining the Hamiltonian H = sp, (2) allows us to write the Hamilton equations q = {q, H} ; t s = {s, H} ; if {q, p} =1, {s, p} = {s, q} =0. (3) t Of course, if we start with a probability istribution P (q, s, 0) at t =0,wehaveasa solution P (q, s, t) =P(q st,s,0), (4) anonecanwritep ψ 2,whereψ is a wave function, also satisfying This ψ obeys the Schröinger equation 1 ψ(q, s, t) =ψ(q st,s,0). (5) t ψ(q, s, t) = iĥψ(q, s, t), if Ĥ = s ˆp, ˆp= i q. (6) Writing things this way, we see that the quantum moel of Eq. 6 is physically an mathematically ientical to the classical one. Here, we write quantum between quotation marks because it can harly be calle a quantum theory in the usual sense: the wave function ψ oes not sprea as time procees (so one cannot envisage interference experiments), an the Hamiltonian Ĥ is not boune from below, so that there is no groun state. It is of importance, however, to emphasize that it is entirely legal to attach to this wave function ψ a Copenhagen probability interpretation. Quantum mechanical superposition of states ψ is permitte in the usual way. Later, we will see how a lower boun in the Hamiltonian of a true quantum theory may arise. A lower boun will be necessary since we wish to have a groun state, or vacuum state. Next, let us take two sets of such variables, q 1, q 2, s 1 an s 2. Again, q i are real numbers an s i are ±1. If the Hamiltonian is taken to be (from now on we omit the hats) H 0 = s 1 p 1 + s 2 p 2, [q i,p j ]=iδ ij ; [s i,s j ]=0, (7) then we have two (istinguishable) particles moving as before. Now, we introuce as interaction s 1 s 1 if q 2 = c, 2

3 where c is a fixe number. This means that particle 1 flips its velocity whenever particle 2 crosses the point c. Clearly, this is a eterministic law. We can express this interaction by means of an extra term in the quantum Hamiltonian: where we introuce the Pauli matrices H = H 0 + H 1 ; H 1 = 1 2 π(σ1 1 1) δ(q 2 c), (8) σ 3 i s i, [σ a i,σb j ]=2iδ ij ε abc σ c i. (9) This Hamiltonian works because q 2 moves with a fixe velocity; the factor 1 2π is exactly what is neee to flip σ1 3 over. The 1 in Eq. 8 is there to ensure that the effect of this Hamiltonian is a flip without a sign change in the wave function: exp ( ( ) iπ(σ1 1 1)) = σ 1 =. (10) 1 0 The positive signs here are important because we wish to fin solutions for ψ that are either positive real numbers or at most have slowly varying phase factors, not epening very much on s i. This will allow us later to construct approximate solutions. The overall sign in the Hamiltonian H 1 is at first sight arbitrary, but this choice turns out to be important later (see remarks near the en of this paper). Next, it will be of essential importance to introuce information loss at the eterministic level: states that are initially ifferent may evolve into the same final state. This is what will make our moel truly quantum mechanical. The resulting quantum theory will still be time-reversible an unitary, because the physical states will be efine to be only those that can be reache in the infinite future 2.Whenq 2 passes a point a, the iscrete variable s 1 willjumpto+1ifitwas 1 before, an if it was +1 it will stay +1. This is escribe by the matrix ( ) 1 1 = (iσ2 + σ 3 + σ 1 +1). (11) Since exp ( iα σ 2 + α(σ 3 + σ 1 1) ) = we can use the interaction Hamiltonian ( ) 1 1 e 2α 0 e 2α, (12) H 2 = α ( σ 2 + i(σ 1 + σ 3 1) ) δ(q 2 a), (13) where α is big, but not too big for our later approximation scheme. Note that H 2 by itself has one real Eigenvalue 0 with Eigenvector ψ(1) = ( 1 0) an a complex one, 3

4 2iαδ(q 2 a), with Eigenvector ψ(2) = 1 2 ( 1 1). In the total Hamiltonian, at a later stage, we will keep only the real Eigenvalues. Similarly, the interaction Hamiltonian H 3 = β ( σ 2 + i(σ 1 σ 3 1) ) δ(q 2 b), (14) with sufficiently large β, will turn the spin s 1 to 1 ifq 2 passes the point b. Again, aitive constants in the Hamiltonians H 2 an H 3 have been chosen such that the transition matrices such as Eq. (11) carry positive signs only. So-far, these Hamiltonians o not look useful for simulating more interesting quantum systems such as harmonic oscillators. But now we concentrate on the many-particle case. Let there be N continuous egrees of freeom q i, i =1,..., N,anN iscrete operators s i = σi 3. Let p ian σi a obey the commutation rules (7) an (9). The Hamiltonian is generalize into H = σi 3 p i + ( 1 2 π(σ1 i 1) δ(q j c k ij )+α( σi 2 +i(σ1 i +σ3 i 1)) δ(q j a k ij )+ i i,j,k +β ( ) σi 2 +i(σ1 i σ3 i 1)) δ(q j b k ij ), (15) The big summation here goes over values of j i, an at each pair (i, j) theremaybe several k values. The points a k ij, bk ij an ck ij are the free parameters of the moel. As must be clear from the foregoing, this moel is entirely eterministic; when q j takes one of the values c k ij, the velocity of the coorinate q i changes sign, if q j = a k ij, this velocity becomes +1, if q j = b k ij, this velocity becomes 1. We are particularly intereste in the case where, for any given j,thepointsc k ij are fairly ensely an smoothly istribute (the points a k ij an bk ij, where information loss takes place, may be very scarce.) We imagine that the eterministic moel will become chaotic. In orer to approximate its behaviour statistically, we construct an approximate solution of the Eigenvalues an Eigenstates of H, using a Hartree-Fock approximation. Take a trial wave function ψ of the form ψ ( {q i,s i } ) = i ψ i (q i,s i ), (16) where ψ i are sufficiently smooth single-particle wave functions. In a more realistic theory, at a later stage, we may be intereste in regaring the qi as fiel values, an replace the inex i by a 3-space coorinate. 4

5 Stanar variation techniques tell us that the best functions ψ i are obtaine as follows. Define 1 2 π ψ j δ(q j c k ij) ψ j = C i ; (17) j,k α ψ j δ(q j a k ij) ψ j = A i ; (18) j,k β ψ j δ(q j b k ij ) ψ j = B i ; (19) j,k ψ i 1 2 πδ(q j c k ij )(σ1 i 1) + α( σi 2 + i(σ1 i + σ3 i 1)) δ(q j a k ij )+ i,k +β ( σ 2 i +i(σ1 i σ3 i 1)) δ(q j b k ij ) ψi = V j (q j )+iw j (q j ). (20) The one-particle wave functions are then seen to obey H i ψ i = E i ψ i ; H i = σi 1 (C i + ia i + ib i )+ +σi 2 (B i A i )+σi 3 (p i+ia i ib i ) C i ia i ib i + V i (q i )+iw i (q i ). (21) If we are only intereste in the Eigenvalues E i, a renormalization of the wave functions allows us to replace p i by ˆp i p i + ia i ib i. (22) The Eigenvalues E i are (for simplicity we rop the inex i): (C ) 2 E = C i(a + B)+V(q)+iW (q) ± + i(a + B) +(B A)2 +ˆp 2 = [ ] = C i(a+b)+v +iw ± C + i(a + B)+ (B A)2 +ˆp 2 2 ( C+i(A+B) )+O(ˆp4 ). (23) Im(E) Re(E) Fig. 1. The complex energy espectrum. Physical states will mostly be within the otte circle. Now, if we ha no information loss, that is, A = B = W = 0, we woul have obtaine a quantum particle with mass ±C. The negative mass values are a serious problem here. 5

6 In eterministic moels, the negative Eigenvalues of the Hamiltonian have always been a serious ifficulty. 1, 3 The most important result we wish to report here is the effect of information issipation. We still assume A an B to be very small compare to C. In this case, the Eigenvalues with positive signs in Eq. (23) are practically real, whereas the others have substantial negative imaginary parts. These must be the states that o not survive at t. The other states have a tiny negative imaginary part. Knowing that there must exist many real Eigenvalues, we imagine that higher orer corrections will be big enough to remove the tiny imaginary parts for many of the lower energy states (insie the otte circle in Figure 1. We fin that only the states with positive energies survive. This sign preference may be explaine as follows. If we ha chosen the signs in Eq. (10) oppositely, we woul have foun that only the negative energy values survive. Thus, the sign choice in Eq. (10) must be seen as establishing the sign convention for the entire Hamiltonian. Corrections to the Hartree-Fock aproximation will give interactions between the particle-like objects. Since their matrix elements are non-vanishing, there will be contributions from virtual negative-energy states, but these will not occur in the final states because they are non-physical. By construction, the exact evolution matrix U = e iht will be unitary within the sector of the physical states, which is why we expect the exact physical spectrum to contain real E values only. Substituting the solutions for ψ i back into Eq. 20, we fin that V an W vanish as soon as B = A, since then ψ i (q i,s i ) = 1. But it is easy to introuce non-trivial bounary conitions for the q s, so that p 2 i =0,an σ1 i 0, in which case non-trivial potentials V an W may emerge (with W V ). Here we observe the effects of the 1 ineq.8: it removes unnecessary large interaction terms so that the Hartree Fock approximation works optimally. There exist of course more ways to generalize the moel. Most essential features seem te be the fact that there are continuous variables q i (so that an infinitesimal time shift can be efine), an iscrete variables s i (allowing us to introuce information loss). Our arguments claiming that non-hartree Fock corrections will prouce a real (an positive) energy spectrum are as yet elicate, an may be calle inconclusive. We suggest that whether or not a non-trivial spectrum of real energy values emerges may epen on further etails of the moel. It will probably be essential that our moel be chaotic, a feature that will epen on etails in a complicate way. An other way to look at the moel is to consier the N limit, an assume that every particle interacts with 6

7 every other one. In this limit, C may ten to infinity an if we keep A an B small, the positive energy values will become real. We have a real quantum theory. Neeless to say, the moel can be generalize in many ways. Here, we merely wante to point out some of its most essential features. References 1. G. t Hooft, J. Stat. Physics 53 (1988) 323; Nucl. Phys. B342 (1990) G. t Hooft, to be publ. 3. G. t Hooft, K. Isler an S. Kalitzin, Nucl. Phys. B386 (1992) 495. See footnote on Page 4 7