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On an alleged “proof” of the quantum probability law
Volume 145, number
PHYSICS
2,3
ON AN ALLEGED
“PROOF”
LETTERS
OF THE QUANTUM
2 April 1990
A
PROBABILITY
LAW
Euan J. SQUIRES Department of Mathematical Science, University of Durham, Durham City, DHI 3LE, UK Received 4 December 1989; revised manuscript Communicated by J.P. Vigier
received
30 January
1990; accepted
for publication
31 January
1990
We endeavour to show that, contrary to many claims in the literature, it is not possible to deduce the quantum probability rule by considering many copies of a system and using the fact that we get a unique result when a system is in an eigenstate of the measured observable.
Most of the successful predictions of quantum theory depend on the assumption that measurement of an observable, say, q, on a system in the normalized state v/yields the result vi with probability 1(vi 1w) I’, where the 1q,) are the normalised eigenstates of V: (1)
919i)=VrI’li).
Usually this is taken as one of the basic axioms of quantum theory. However, beginning with the work of Everett [ 11, there have been many attempts to derive the result from the weaker assumption that if the system is in an eigenstate of a particular Hermitian operator, then measurement of the corresponding observable will always yield the corresponding eigenvalue (see, for example, refs. [ 2-61). If such a derivation was valid then it would do more than merely simplify the axioms of quantum theory; in addition it would seem to remove one of the major problems of the many-worlds interpretation of quantum theory, namely how can we understand the probability law when nothing beyond the unitary evolution described by the Schrodinger equation actually happens? There just is not anything of which I ( q, I y) I ’ can be the probability (see refs. [ 7-101). In this note, however, we shall endeavour to show that this particular method of rescuing the many-worlds idea is not available. The mathematics does not support the claimed physical result. To be explicit we consider an operator with two eigenvalues 0375-9601/90/$
03.50 0 Elsevier
Science Publishers
Oln)=flln)
>
(2)
where n = 0, 1. For example we might take O= 4 + o,, for a spin-t particle. Then we can expand the normalised state of a given system, 01, in the form
lva)=cll)a+~lo),>
(3)
where c2+s2= 1 and, for simplicity, we take c and s to be real. The wavefunction associated with N copies of this state is given by
lW=
n Iwcr>.
(4)
a
We also define an “average”
operator
Q=;;o,.
Q by
(5)
The mean value of Q in the state I Y) is then easily seen to be given by Q=(YlQl!P)=c2.
(6)
The point at issue now is to what extent we can regard I Y) as being “nearly” an eigenstate of Q with eigenvalue c2. If we write
QlW=c21V+lx>
>
(7)
then we have
(xlx>=(~lQ21~>-2c2(~lQlY’>+~4
B.V. (North-Holland
= (c4-c’/N+c’/N)-2c4+c4 =c2s2/N. )
(8) 67
Volume
145, number
PHYSICS
2,3
It follows that (XIX)-0
as
N--ra~.
(9)
Thus, in this sense, we could say that 1Y) becomes an eigenstate of Q in the limit that N+cc. There are, however, several problems in using this purely mathematical argument to deduce anything of relevance to physics. First, it is clear that for no value of N is 1Y) actually an eigenstate of Q. Fahri et al. [6] endeavour to overcome this problem by defining the state 1!P) Secondly, it is not obvious that there is any reason why we should be interested in any particular scalar product as a measure of the deviation of one state from another. The relevance of the scalar product is very much connected with the probability law, so we are in danger of having assumed at least a part of our claimed result. Finally, as we shall see immediately, far from being close to the eigenstate of Q, the state 1Y) is actually orthogonal to it in the limit when N-co. To establish this we note that the eigenstates of Q with eigenvalue c2 are given by
(10) where p labels a particular ordering. Of course this requires Nc2 to be an integer, otherwise there are no exact eigenstates with this eigenvalue. Clearly then ( YYJY; ) = cNc2PZ .
(11)
Thus, except in the uninteresting case when one of c or s is equal to unity, the state ( Y) is orthogonal to these eigenstates in the large N limit, for any value of p. Since we are interested in the limit when N becomes infinite this is not yet quite enough, however. We must consider the particular combination for which the projection is maximum. This is for the symmetrical state: 1 YE)
=c-“2
c
1 YF>
(12)
)
P
where C is the number
68
of orderings
possible,
i.e.
LETTERS
2 April 1990
A
N! (Ns')!
“= (NC')!
(13)
.
Then ( y] yE) = Cr12cN@sNs2
(14)
We can evaluate this in the large N limit by means of Stirling’s formula, which gives ( Y] Y’E) =
1 (2r~N)“~(cs)“~
’
(15)
which tends to zero in the limit of infinite N. This surely means that we cannot deduce anything about the properties of the state I Y) with regard to the observable I], from statements about properties of the eigenstates of the observable. The claim that we can eliminate the basic probability assumption of quantum theory is therefore false. As noted earlier, this discussion is relevant to the many-worlds interpretation of quantum theory. As I have already claimed [ 7,8], this model is not consistent with the world as we observe it, unless some additional assumptions, involving something that “happens”, are added. Similar points have been made by Stein [9], Ben-Dov [ll] and Kent [lo].
References 11 ] H. Everett, Rev. Mod. Phys. 29 ( 1957) 454. [I21 N. Graham, in: The many-worlds interpretation
of quantum mechanics, eds. B. Dewitt and N. Graham (Princeton Univ. Press, Princeton, 1973). 31 B. Dewitt, in: The many-worlds interpretation of quantum mechanics, eds. B. De Witt and N. Graham (Princeton Univ. Press, Princeton, 1973). [4] J.B. Hartle, Am. J. Phys. 36 ( 1968) 704. [5]R.Geroch,NoDs18 (1984) 617. [ 61 E. Fahri, J. Goldstone and S. Gutmann, Ann. Phys. (NY) 192 (1989) 368. [7] E.J. Squires, Found. Phys. Lett. 1 ( 1988) 13. [ 81 E.J. Squires, in: Proc. Conf. on Quantum theory without collapse (1989), eds. M. Cini and J.M. Levy-Leblond (Hilger, Bristol), to be published. [9] H. Stein, Nous 18 (1984) 635. [ 10 ] A. Kent, Against many-worlds interpretations, Princeton preprint IASSNS-HEP-89/36. [ I I ] Y. Ben-Dov, 62 years of uncertainty, talk at Erice conference (1989).
PHYSICS
2,3
ON AN ALLEGED
“PROOF”
LETTERS
OF THE QUANTUM
2 April 1990
A
PROBABILITY
LAW
Euan J. SQUIRES Department of Mathematical Science, University of Durham, Durham City, DHI 3LE, UK Received 4 December 1989; revised manuscript Communicated by J.P. Vigier
received
30 January
1990; accepted
for publication
31 January
1990
We endeavour to show that, contrary to many claims in the literature, it is not possible to deduce the quantum probability rule by considering many copies of a system and using the fact that we get a unique result when a system is in an eigenstate of the measured observable.
Most of the successful predictions of quantum theory depend on the assumption that measurement of an observable, say, q, on a system in the normalized state v/yields the result vi with probability 1(vi 1w) I’, where the 1q,) are the normalised eigenstates of V: (1)
919i)=VrI’li).
Usually this is taken as one of the basic axioms of quantum theory. However, beginning with the work of Everett [ 11, there have been many attempts to derive the result from the weaker assumption that if the system is in an eigenstate of a particular Hermitian operator, then measurement of the corresponding observable will always yield the corresponding eigenvalue (see, for example, refs. [ 2-61). If such a derivation was valid then it would do more than merely simplify the axioms of quantum theory; in addition it would seem to remove one of the major problems of the many-worlds interpretation of quantum theory, namely how can we understand the probability law when nothing beyond the unitary evolution described by the Schrodinger equation actually happens? There just is not anything of which I ( q, I y) I ’ can be the probability (see refs. [ 7-101). In this note, however, we shall endeavour to show that this particular method of rescuing the many-worlds idea is not available. The mathematics does not support the claimed physical result. To be explicit we consider an operator with two eigenvalues 0375-9601/90/$
03.50 0 Elsevier
Science Publishers
Oln)=flln)
>
(2)
where n = 0, 1. For example we might take O= 4 + o,, for a spin-t particle. Then we can expand the normalised state of a given system, 01, in the form
lva)=cll)a+~lo),>
(3)
where c2+s2= 1 and, for simplicity, we take c and s to be real. The wavefunction associated with N copies of this state is given by
lW=
n Iwcr>.
(4)
a
We also define an “average”
operator
Q=;;o,.
Q by
(5)
The mean value of Q in the state I Y) is then easily seen to be given by Q=(YlQl!P)=c2.
(6)
The point at issue now is to what extent we can regard I Y) as being “nearly” an eigenstate of Q with eigenvalue c2. If we write
QlW=c21V+lx>
>
(7)
then we have
(xlx>=(~lQ21~>-2c2(~lQlY’>+~4
B.V. (North-Holland
= (c4-c’/N+c’/N)-2c4+c4 =c2s2/N. )
(8) 67
Volume
145, number
PHYSICS
2,3
It follows that (XIX)-0
as
N--ra~.
(9)
Thus, in this sense, we could say that 1Y) becomes an eigenstate of Q in the limit that N+cc. There are, however, several problems in using this purely mathematical argument to deduce anything of relevance to physics. First, it is clear that for no value of N is 1Y) actually an eigenstate of Q. Fahri et al. [6] endeavour to overcome this problem by defining the state 1!P) Secondly, it is not obvious that there is any reason why we should be interested in any particular scalar product as a measure of the deviation of one state from another. The relevance of the scalar product is very much connected with the probability law, so we are in danger of having assumed at least a part of our claimed result. Finally, as we shall see immediately, far from being close to the eigenstate of Q, the state 1Y) is actually orthogonal to it in the limit when N-co. To establish this we note that the eigenstates of Q with eigenvalue c2 are given by
(10) where p labels a particular ordering. Of course this requires Nc2 to be an integer, otherwise there are no exact eigenstates with this eigenvalue. Clearly then ( YYJY; ) = cNc2PZ .
(11)
Thus, except in the uninteresting case when one of c or s is equal to unity, the state ( Y) is orthogonal to these eigenstates in the large N limit, for any value of p. Since we are interested in the limit when N becomes infinite this is not yet quite enough, however. We must consider the particular combination for which the projection is maximum. This is for the symmetrical state: 1 YE)
=c-“2
c
1 YF>
(12)
)
P
where C is the number
68
of orderings
possible,
i.e.
LETTERS
2 April 1990
A
N! (Ns')!
“= (NC')!
(13)
.
Then ( y] yE) = Cr12cN@sNs2
(14)
We can evaluate this in the large N limit by means of Stirling’s formula, which gives ( Y] Y’E) =
1 (2r~N)“~(cs)“~
’
(15)
which tends to zero in the limit of infinite N. This surely means that we cannot deduce anything about the properties of the state I Y) with regard to the observable I], from statements about properties of the eigenstates of the observable. The claim that we can eliminate the basic probability assumption of quantum theory is therefore false. As noted earlier, this discussion is relevant to the many-worlds interpretation of quantum theory. As I have already claimed [ 7,8], this model is not consistent with the world as we observe it, unless some additional assumptions, involving something that “happens”, are added. Similar points have been made by Stein [9], Ben-Dov [ll] and Kent [lo].
References 11 ] H. Everett, Rev. Mod. Phys. 29 ( 1957) 454. [I21 N. Graham, in: The many-worlds interpretation
of quantum mechanics, eds. B. Dewitt and N. Graham (Princeton Univ. Press, Princeton, 1973). 31 B. Dewitt, in: The many-worlds interpretation of quantum mechanics, eds. B. De Witt and N. Graham (Princeton Univ. Press, Princeton, 1973). [4] J.B. Hartle, Am. J. Phys. 36 ( 1968) 704. [5]R.Geroch,NoDs18 (1984) 617. [ 61 E. Fahri, J. Goldstone and S. Gutmann, Ann. Phys. (NY) 192 (1989) 368. [7] E.J. Squires, Found. Phys. Lett. 1 ( 1988) 13. [ 81 E.J. Squires, in: Proc. Conf. on Quantum theory without collapse (1989), eds. M. Cini and J.M. Levy-Leblond (Hilger, Bristol), to be published. [9] H. Stein, Nous 18 (1984) 635. [ 10 ] A. Kent, Against many-worlds interpretations, Princeton preprint IASSNS-HEP-89/36. [ I I ] Y. Ben-Dov, 62 years of uncertainty, talk at Erice conference (1989).