Download Relating Probability Amplitude Mechanics to

Relating Probability Amplitude Mechanics to
Standard Statistical Models
Robert F. Bordley
June 15, 2006
Abstract
The probabilities assessed for an event by a detailed experiment (e.g.,
one which measures both a particle's position on a detection screen and
which of several slits it traversed to reach that screen) may dier from the
probabilities assessed for that same event by a less detailed experiment
(e.g., one which only measures the particle's position on the detection
screen.) As this paper shows, probability amplitude mechanics writes the
detailed experiment probabilities as a weighted average of the less detailed experiment probabilities and some factor measuring the variability
overlooked by the less detailed experiment.
1 MOTIVATION
The n-slit interference experiments show that the probability assessed for a
specic event, e.g., the particle reaching a specic point on a detector screen,
will vary depending upon whether the experiment measuring that event was
detailed (e.g., also assessed which slit the particle traversed en route to the
screen) or less detailed (only assessed the particle's position on the detector
screen). This paper will show that probability amplitude mechanics implies
that the probability for an event, given a detailed experiment, is a weighted
average of
the probability of that same event, given a less detailed experiment, and
a factor measuring the variability observed by the detailed experiment but
overlooked by the less detailed experiment.
We then show that this alternate representation of probability amplitude mechanics might be deducible from fairly simple statistical assumptions. This
Operating Sciences Department; General Motors Research Labs; Warren, Michigan 480909055
1
suggests the possibility of deriving probability amplitude mechanics from intuitive rst principles, a result which would help resolve interpretational issues
associated with quantum theory1 .
2 THE WEIGHTED AVERAGE FORMULA
2.1
More Detailed and Less Detailed Experiments
Let be the universal set and ; be the null set. Following operational statistics(Foulis & Randall,1972a,1972b), we dene the set of possible outcomes resulting from applying a specic operation C to a system by the partition
C
= (Aj[A = ; Ai \ Aj = ;; i 6= j )
Now consider a second operation F whose possible outcomes are represented by
the partition
F = (E j[E = ; Ei \ Ej = ;; i 6= j )
If every possible outcome of F is a subset of every possible outcome of C , i.e.,
if E 2 F implies there exists an A 2 C with E A, then operation C is less
detailed (or more `coarse') than operation F .
For operational statistics, the probability of observing event A, given we
perform operation C , is Pr(Aj C ), the probability assessed for the state A given
an operation whose possible outcomes are represented by the elements of C .
Now consider the probability for this same state A if we planned to perform the
more detailed operation F instead of C , i.e., Pr(Aj F ))2 .
2.2
Probability Amplitude Mechanics
To determine this probability, dene the variance of event A as the sum of
the variance of the real and imaginary wave function components of each event
E 2 A. Then the next section proves:
Theorem: Probability Amplitude Mechanics implies
Pr(Aj
F)
= (1
j
) Pr(A C ) + VA
r
1+r and r is the coecient of variation of events E 2 A,
averaged over all A 2 C . Thus r measures the variability overlooked by
the less detailed experiment (and 0 1:)
where =
1 When asked to explain probability amplitude mechanics, Feynman(1965) had replied, \We
have no ideas about a more basic mechanism from which these results can be deduced."
2 The fact that these two probabilities are dierent in the case of the n-slit interference
experiment is, of course, one of the fundamental `anomalies' of quantum mechanics.
2
where
VA is the variance of event A divided by the summed variance
of all events A 2 C . The probability distribution, VA , assigns higher
probability to more variable states | somewhat like the maximum entropy
principle.
When r is zero, there is no variability within the states, A, and everything the
more detailed experiment measures is consistent with what the less detailed
experiment had measured. In this case, Pr(Aj F ) = Pr(Aj C ), i.e., the detailed
and less detailed experiment assign the same state probabilities. When r is
innite, the more detailed experiment measures an innitely large amount of
variability not measured by the less detailed experiment. Hence the results
of the less detailed experiment are essentially irrelevant and the probabilities
assigned to various states are determined by VA . When r is between zero and
innity, the detailed experiment's probability is intermediate between the less
detailed experiment's probability and VA . This, of course, is quite intuitive.
To write the formula in a somewhat more traditional matter, note that operational statistics mandates Pr(Aj F ) = E 2A Pr(E j F ), i.e., once we condition
our probabilities on the same experiment, all the standard rules of probability
apply. Making this substitution and rearranging the results of the Theorem
gives
P
Pr(Aj
C)
= Pr(Aj
F )+
1
(Pr(Aj
F ) VA )
P
=
X Pr(Ej
E 2A
F )+r (Pr(A
j F)
VA )
Hence the traditional interference term, ( E;E 2A [Pr(E j F ) Pr(E j F )]:5 cos(E
E ), has been replaced by r(Pr(Aj F ) VA ). Note that as the amount of
unmeasured variability, r goes to zero, probability amplitude mechanics gives
additive probabilities. Hence we get a simple correspondence principle between
classical and quantum mechanics.
2.3
Derivation from a Simple Statistical Model
The fact that the Theorem is so intuitive suggests that the same formula might
be deducible from simpler statistical models. As a step toward constructing
such a derivation, note that the probabilities assessed by the more detailed experiment, Pr(Aj F ), will be a function of those aspects of the system measured
by the less detailed experiment (and reected in Pr(Aj C )), and some aspects
unmeasured by the less detailed experiment (reected in some noise term.) The
simplest statistical assumption about this noise term (given that it represents
uncertainty about a probability) is that it follows a Dirichlet distribution. We
similarly follow statistical mechanics in assuming that Pr(Aj C ) represents the
mode (i.e., the most likely value) of this Dirichlet distribution.
Given these assumptions, the Appendix deduces Pr(Aj F ) as the mean of
this distribution. We state this as a Lemma:
Lemma:Pr(Aj F ) = (1
) Pr(Aj C ) + j 1 j where 0 1.
C
3
which is identical to the result of the Theorem when each event has the same
variance, i.e., VA = j 1 j .
This special case does involve an interference term, r(Pr(Aj F ) j 1 j ). But
since Pr(Aj C ) > E 2A Pr(E j F ) if and only if Pr(Aj F ) > VA , the case of
VA = j 1 j implies that whenever Pr(Aj F ) is larger then average, Pr(Aj C ) >
Pr(Aj F ) and whenever Pr(Aj F ) is smaller than average, Pr(Aj C ) < Pr(Aj C ).
Thus the Dirichlet special case implies that the probabilities from the less detailed experiment tend to be more extreme than the probabilities from the
detailed experiment. Since this is not true in many quantum mechanical cases,
future work will have to specify a distribution more general than the Dirichlet.
P
C
C
C
3 PROOF OF THE THEOREM
3.1
An Alternate Way of Writing the Wave Function
Probability amplitude mechanics writes the probability of
wave functions, , as
A A
Pr(Aj C ) / A A =
A A
A2
P
with
A
=
X
E 2A
A
in terms of the
(1)
C
(2)
E
While the wave function, E , is commonly written in the form E = (Pr(E )):5 exp(iE ),
this paper writes it as E = mE;R + imE;I . Given this representation3 , (1) is
equivalent to
Pr(Aj
C ) / [mA;R + imA;I ][mA;R
imA;I ] =
2
2
P 2m (m2+ m+ m2
A;R
A
A;I
A;R
C
A;I )
(3)
For simplicity, we let Z be a dummy index which can either equal R or I .
3.2
The Variances associated with Wave Functions
For both values of Z , dene the variances of the real and imaginary parts of the
wave functions associated with events in A by
vA;Z
=
1
X
(mE;R
2jAj E;E 2A
mE ;R )2
=
X m2
E 2A
E;R
1
(
Xm
jAj E2A
E;R )
2
(4)
For Bohm(1952), E = ShE which implies that mE;R = (Pr(E )):5 cos(SE =h) and mE;I =
(Pr(E )):5 sin(SE =h) so that both m-functions can be negative.
3
4
These two variances, vA;R and vA;I measure the amount of variability `internal'
to the event A and which, presumably, would be overlooked by the less detailed
experiment whose only outcomes are A 2 C .
Equation (2) is equivalent to
mA;Z
=
Xm
E;Z f or Z
E 2A
= R; I
(5)
Substituting (5) in (4) gives, with some rearrangement,
m2A;Z
Dening DC =
P2
A
C
= jAj[
X m2
2
;Z mA;Z
Pr(Aj
C)
=
vA;Z ] f or Z
E;Z
E 2A
= R; I
(6)
and substituting (6) in (3) gives
jAj[PZ;E2A m2E;Z
P
Z vA;Z ]
(7)
DC
Now consider an operation F which does measure the probability of each elemental event E 2 A. Then the analogue of (3) gives
Dening DF =
Pr(E j
P
Z;E 2
Pr(Aj
F
m2E;Z ,
C ) = jAj
F) =
P
P
2
Z mE;Z
Z;E 2
F
(8)
m2E;Z
and substituting (8) in (7) gives
P2
E A Pr(E
Pv
j F )DF
z A;Z
DC
(9)
As a comparison, recall that writing the wave function as E = (Pr(E )):5 exp(iE )
leads to the formula
Pr(Aj C ) =
Pr(E j F ) +
(Pr(E j F ) Pr(E j F )):5 cos(E E )
E 2A
E;E 2A
X
X
The key dierence between the two formulas is that (9) uses variances to measure interference eects. Since interference eects are alien to standard statistics
while variances are commonplace, this change is critical to translating probability amplitude mechanics into the vocabulary of standard statistics.
3.3
Comparing Detailed and Less Detailed Experiments
P
Operational statistics indicates that all probabilities derived from the same experiment obey the standard rules of probability. Hence Pr(Aj F ) = E 2A Pr(E j
Making this substitution in (9) gives
Pr(Aj
C ) = j Aj
Pr(Aj
5
F )DF
DC
P
Z vA;Z
(10)
F ).
Note that
=
DF
=
=
X
Z;E 2
m2E;Z
X X m2
=
Z;A2
X [ X m2
Z;A2
X
Z;A2
F
vA;Z
Pr(Aj
C)
+ DC
P
P
=
E;Z
m2A;Z ] +
Z;A2
Z;A2
C
so that
1
E 2A
j Aj
E;Z
E 2A
C
C
C
C
Z;A2
m2A;Z
jAj
m2
X
A;Z
=
m2A;Z
C
j Aj
X
Z;A2
vA;Z
+ DC
C
jAj [Pr(Aj )(X v + D X Pr(Aj C ) )
F
A;Z
C
D
jAj
C
Z;A
A
X
A2
Pr(Aj
jAj
C
C)
Xv
Z
A;Z ]
We can rewrite this expression as
Pr(Aj
with
j
Q(A C )
VA
=
=
=
F)
j
= (1
)Q(A C ) + VA
PPr(Pr(AjAj )=j)A=jjAj
P
P vv
P v
P Pr(Aj )=jAj + P
(11)
C
C
A
Z A;Z
A;Z A;Z
Z;A A;Z
z
Z;A vA
C
A
P
=P
Z;A vA;Z
2
Z;E mE;Z
Thus the probability of event A as assessed by the detailed operation, F , is a
weighted average of
the probability of event A as assessed by the less detailed operation, adjusted (via Q(Aj C )), to increase that probability if A is more detailed
(i.e., contains fewer elements of F ) than other sets A 2 C .
the variance of elemental m-functions within set A divided by that variance
summed across all A 2 C .
To understand , dene MA;Z to be a random variable assuming values mE;Z
for each E 2 A with probability jA1 j . The mean of this random variable is
m
m
2
jAj . The mean squared is ( jAj ) . The variance of this random variable is
2
m
m
v
2
( 2jAj
)2 . Hence the coecient of variation, rA;Z , of
jAj =
jA j
this random variable, is
vA;Z
rA;Z = 2
A;Z
A;Z
P
E
A
E;Z
P
E
A;Z
A
E;Z
j j
mA;Z = A
6
The average coecient of variation across all A 2
r
Then
=
P
P
=
P
P
A;Z vA;Z
EZ
A;Z = A
A;Z
AZ vA;Z
E;Z
Xr P
=
A;Z
m2
A;Z
A;Z
P
+
AZ vA;Z
AZ vA;Z
AZ
is
j j
A;Z = A
P
P
=
m2
j j
m2
C
m2
j j
m2
j j
A;Z = A
A;Z = A
=
r
r+1
For many physical applications, jAj can be taken as constant across all elements of C . In this case, (11) simplies to
Pr(Aj
F)
j
= (1
) Pr(A C ) + VA
with =
which proves the Theorem4 .
r
r+1
APPENDIX:THE DIRICHLET DENSITY
If random variables MA2 and MA2 both have generalized Rayleigh (or noncentral Chi Square) distributions(Park,1961) with 2A and 2A degrees of free2
dom and non-centrality factors, A and A , then the ratio pA = m2 m+m2 has
a non-central beta distribution. Because of the analytical complexity of the
non-central beta distribution, we focus on the simpler beta distribution (which
implies A = A = 0 with MA2 and MA2 having Chi Square Distributions.)
The multivariate generalization of the beta distribution is the Dirichlet density(Johnson & Kotz,1970):
A
A
pf (pA ; pA ; :::; pA )
/
Y
A2
pAA 1 (1
C
with the expected value of pA given by
E [pA ] =
P 2
Xp
A2
!0
A)
1
C
A
A
C
A
and the mode of pA (for A 1 for A 2 C ) given by
A 1
A 1
M [pA ] =
=
(
1)
A j C j
A
A2
A2
Then
j C j )M [ p ] + j C j 1
E [pA ] = (1
A
A
A j C j
A 2
A2
j
j
Substituting =
gives the Lemma. Since A 1; A
2
conclude that 0 1.
P
P
P
=
A
4
Note that VA
P
P
C
(i;j )2A
C
P
C
C
C
A
A
2
.
Vij A
7
C
2
C,
we
References
[1] Bohm, D. \A Suggested Interpretation of the Quantum Theory in terms of
`Hidden Variables'." Physical Review. 85,160-193(1952).
[2] Feynman, R., R. Leighton & M. Sands.
Addison-Wesley, New York, 1965.
Feynman Lectures on Physics.
[3] Foulis, D. & C. Randall. "Operational Statistics, I. Basic Concepts." Journal
of Mathematical Physics. 13, 1667-1675(1972).
[4] Foulis, D. & C. Randall. "Operational Statistics, II. Manuals of Operations
& Their Logics." Journal of Mathematical Physics. 13, 1667-1675(1972).
[5] Johnson, N. & S. Kotz. Continuous
Miin Company, Boston,1970.
. Houghton
Univariate Distributions-2
[6] Park, J. \Moments of the Generalized Rayleigh Distribution." Quarterly of
Applied Mathematics. 19,45-49(1961).
8