Griffiths R.B. Consistent Quantum Theory
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114 Stochastic histories
the quantum history
Y = F
1
( F
2
(8.12)
given by (8.11) is a sum of three orthogonal projectors,
˜
Y = F
1
(
˜
F
2
+
˜
F
1
( F
2
+
˜
F
1
(
˜
F
2
, (8.13)
where
˜
F
j
means I − F
j
. Note that the compound history
˜
Y in (8.13) cannot be
written in the form G
1
( G
2
, that is, as an event at t
1
followed by another event at
t
2
.
The conjunction Y AND Y
,orY ∧ Y
, of two histories is represented by the
product YY
of the projectors, provided they commute with each other. If YY
=
Y
Y, the conjunction is not defined. The situation is thus entirely analogous to the
conjunction of two quantum properties at a single time, as discussed in Secs. 4.5
and 4.6. Let us suppose that the history
Y
= F
1
( F
2
( F
3
(8.14)
is defined at the same three times as Y in (8.7). Their conjunction is represented
by the projector
Y
∧ Y = Y
Y = F
1
F
1
( F
2
F
2
( F
3
F
3
, (8.15)
which is equal to YY
provided that at each of the three times the projectors in the
two histories commute:
F
j
F
j
= F
j
F
j
for j = 1, 2, 3. (8.16)
However, there is a case in which Y and Y
commute even if some of the condi-
tions in (8.16) are not satisfied. It occurs when the product of the two projectors at
one of the times is 0, for this means that YY
= 0 independent of what projectors
occur at other times. Here is an example involving a spin-half particle:
Y = [x
+
] ( [x
+
] ( [z
+
],
Y
= [y
+
] ( [z
+
] ( [z
−
].
(8.17)
The two projectors at t
1
,[x
+
] and [y
+
], clearly do not commute with each other,
and the same is true at time t
2
. However, the projectors at t
3
are orthogonal, and
thus YY
= 0 = Y
Y.
A simple example of a nonvanishing conjunction is provided by a spin-half par-
ticle and two histories
Y = [z
+
] ( I, Y
= I ( [x
+
], (8.18)
8.4 Extensions and logical operations on histories 115
defined at the times t
1
and t
2
. The conjunction is
Y
∧ Y = Y
Y = YY
= [z
+
] ( [x
+
], (8.19)
and this is sensible, for the intuitive significance of (8.19) is “S
z
=+1/2att
1
and
S
x
=+1/2att
2
.” Indeed, any history of the form (8.6) can be understood as “F
1
at t
1
, and F
2
at t
2
, and ... F
f
at t
f
.” This example also shows how to generate
the conjunction of two histories defined at different sets of times. First one must
extend each history by including I at additional times until the extended histories
are defined on a common set of times. If the extended projectors commute with
each other, the operator product of the projectors, as in (8.15), is the projector for
the conjunction of the two histories.
The disjunction “Y
or Y or both” of two histories is represented by a projector
Y
∨ Y = Y
+ Y − Y
Y (8.20)
provided Y
Y = YY
; otherwise it is undefined. The intuitive significance of the
disjunction of two (possibly compound) histories is what one would expect, though
there is a subtlety associated with the quantum disjunction which does not arise in
the case of classical histories, as has already been noted in Sec. 4.5 for the case
of properties at a single time. It can best be illustrated by means of an explicit
example. For a spin-half particle, define the two histories
Y = [z
+
] ( [x
+
], Y
= [z
+
] ( [x
−
]. (8.21)
The projector for the disjunction is
Y ∨ Y
= Y + Y
= [z
+
] ( I, (8.22)
since in this case YY
= 0. The projector Y∨Y
tells us nothing at all about the spin
of the particle at the second time: in and of itself it does not imply that S
x
=+1/2
or S
x
=−1/2att
2
, since the subspace of
˘
H on which it projects contains, among
others, the history [z
+
] ( [y
+
], which is incompatible with S
x
having any value
at all at t
2
. On the other hand, when the projector Y ∨ Y
occurs in the context
of a discussion in which both Y and Y
make sense, it can be safely interpreted as
meaning (or implying) that at t
2
either S
x
=+1/2orS
x
=−1/2, since any other
possibility, such as S
y
=+1/2, would be incompatible with Y and Y
.
This example illustrates an important principle of quantum reasoning: The con-
text, that is, the sample space or event algebra used for constructing a quantum
description or discussing the histories of a quantum system, can make a difference
in how one understands or interprets various symbols. In quantum theory it is
important to be clear about precisely what sample space is being used.
116 Stochastic histories
8.5 Sample spaces and families of histories
As discussed in Sec. 5.2, a sample space for a quantum system at a single time
is a decomposition of the identity operator for the Hilbert space H : a collection
of mutually orthogonal projectors which sum to I. In the same way, a sample
space of histories is a decomposition of the identity on the history Hilbert space
˘
H,
a collection {Y
α
} of mutually orthogonal projectors representing histories which
sum to the history identity:
˘
I =
α
Y
α
. (8.23)
It is convenient to label the history projectors with a superscript in order to be able
to reserve the subscript position for time. Since the square of a projector is equal
to itself, we will not need to use superscripts on projectors as exponents.
Associated with a sample space of histories is a Boolean “event” algebra, called
a family of histories, consisting of projectors of the form
Y =
α
π
α
Y
α
, (8.24)
with each π
α
equal to 0 or 1, as in (5.12). Histories which are members of the sam-
ple space will be called elementary histories, whereas those of the form (8.24) with
two or more π
α
equal to 1 are compound histories. The term “family of histories”
is also used to denote the sample space of histories which generates a particular
Boolean algebra. Given the intimate connection between the sample space and the
corresponding algebra, this double usage is unlikely to cause confusion.
The simplest way to introduce a history sample space is to use a product of
sample spaces as that term was defined in Sec. 6.6. Assume that at each time t
j
there is a decomposition of the identity I
j
for the Hilbert space H
j
,
I
j
=
α
j
P
α
j
j
, (8.25)
where the subscript j labels the time, and the superscript α
j
labels the different
projectors which occur in the decomposition at this time. The decompositions
(8.25) for different values of j could be the same or they could be different; they
need have no relationship to one another. (Note that the sample spaces for the
different classical systems discussed in Sec. 8.2 have this sort of product structure.)
Projectors of the form
Y
α
= P
α
1
1
( P
α
2
2
(···P
α
f
f
, (8.26)
where α is an f -component label
α = (α
1
,α
2
,...α
f
), (8.27)
8.5 Sample spaces and families of histories 117
make up the sample space, and it is straightforward to check that (8.23) is satisfied.
Here is a simple example for a spin-half particle with f = 2:
I
1
= [z
+
] + [z
−
], I
2
= [x
+
] + [x
−
]. (8.28)
The product of sample spaces consists of the four histories
Y
++
= [z
+
] ( [x
+
], Y
+−
= [z
+
] ( [x
−
],
Y
−+
= [z
−
] ( [x
+
], Y
−−
= [z
−
] ( [x
−
],
(8.29)
in an obvious notation. The Boolean algebra or family of histories contains 2
4
=
16 elementary and compound histories, including the null history 0 (which never
occurs).
Another type of sample space that arises quite often in practice consists of histo-
ries which begin at an initial time t
0
with a specific state represented by a projector
"
0
, but behave in different ways at later times. We shall refer to it as a family based
upon the initial state "
0
. A relatively simple version is that in which the histories
are of the form
Y
α
= "
0
( P
α
1
1
( P
α
2
2
(···P
α
f
f
, (8.30)
with the projectors at times later than t
0
drawn from decompositions of the identity
of the type (8.25). The sum over α of the projectors in (8.30) is equal to "
0
,soin
order to complete the sample space one adds one more history
Z = (I − "
0
) ( I ( I (···I (8.31)
to the collection. If, as is usually the case, one is only interested in the histories
which begin with the initial state "
0
, the history Z is assigned zero probability,
after which it can be ignored. The procedure for assigning probabilities to the
other histories will be discussed in later chapters. Note that histories of the form
(I −"
0
) ( P
α
1
1
( P
α
2
2
(···P
α
f
f
(8.32)
are not present in the sample space, and for this reason the family of histories based
upon an initial state "
0
is distinct from a product of sample spaces in which (8.25)
is supplemented with an additional decomposition
I
0
= "
0
+ (I − "
0
) (8.33)
at time t
0
. As a consequence, later events in a family based upon an initial state "
0
are dependent upon the initial state in the technical sense discussed in Ch. 14.
Other examples of sample spaces which are not products of sample spaces are
used in various applications of quantum theory, and some of them will be discussed
in later chapters. In all cases the individual histories in the sample space correspond
to product projectors on the history space
˘
H regarded as a tensor product of Hilbert
118 Stochastic histories
spaces at different times, (8.5). That is, they are of the form (8.6): a quantum
property at t
1
, another quantum property at t
2
, and so forth. Since the history space
˘
H is a Hilbert space, it also contains subspaces which are not of this form, but
might be said to be “entangled in time”. For example, in the case of a spin-half
particle and two times t
1
and t
2
, the ket
|#=
|z
+
(|z
−
−|z
−
(|z
+
/
√
2 (8.34)
is an element of
˘
H, and therefore [#] =|##| is a projector on
˘
H. It seems
difficult to find a physical interpretation for histories of this sort, or sample spaces
containing such histories.
8.6 Refinements of histories
The process of refining a sample space in which coarse projectors are replaced with
finer projectors on subspaces of lower dimensionality was discussed in Sec. 5.3.
Refinement is often used to construct sample spaces of histories, as was noted in
connection with the classical random walk in one dimension in Sec. 8.2. Here is
a simple example to show how this process works for a quantum system. Con-
sider a spin-half particle and a decomposition of the identity {[z
+
], [z
−
]} at time
t
1
. Each projector corresponds to a single-time history which can be extended to a
second time t
2
in the manner indicated in Sec. 8.3, to make a history sample space
containing
[z
+
] ( I, [z
−
] ( I. (8.35)
If one uses this sample space, there is nothing one can say about the spin of the
particle at the second time t
2
, since I is always true, and is thus completely un-
informative. However, the first projector in (8.35) is the sum of [z
+
] ( [z
+
] and
[z
+
] ( [z
−
], and if one replaces it with these two projectors, and the second pro-
jector in (8.35) with the corresponding pair [z
−
] ( [z
+
] and [z
−
] ( [z
−
], the result
is a sample space
[z
+
] ( [z
+
], [z
+
] ( [z
−
],
[z
−
] ( [z
+
], [z
−
] ( [z
−
],
(8.36)
which is a refinement of (8.35), and permits one to say something about the spin at
time t
2
as well as at t
1
.
When it is possible to refine a sample space in this way, there are always a
large number of ways of doing it. Thus the four histories in (8.29) also constitute
arefinement of (8.35). However, the refinements (8.29) and (8.36) are mutually
incompatible, since it makes no sense to talk about S
x
at t
2
at the same time that
one is ascribing values to S
z
, and vice versa. Both (8.29) and (8.36) are products
8.7 Unitary histories 119
of sample spaces, but refinements of (8.35) which are not of this type are also
possible; for example,
[z
+
] ( [z
+
], [z
+
] ( [z
−
],
[z
−
] ( [x
+
], [z
−
] ( [x
−
],
(8.37)
where the decomposition of the identity used at t
2
is different depending upon
which event occurs at t
1
.
The process of refinement can continue by first extending the histories in (8.36)
or (8.37) to an additional time, either later than t
2
or earlier than t
1
or between t
1
and t
2
, and then replacing the identity I at this additional time with two projec-
tors onto pure states. Note that the process of extension does not by itself lead
to a refinement of the sample space, since it leaves the number of histories and
their intuitive interpretation unchanged; refinement occurs when I is replaced with
projectors on lower-dimensional spaces.
It is important to notice that refinement is not some sort of physical process
which occurs in the quantum system described by these histories. Instead, it is
a conceptual process carried out by the quantum physicist in the process of con-
structing a suitable mathematical description of the time dependence of a quantum
system. Unlike deterministic classical mechanics, in which the state of a system at
a single time yields a unique description (orbit in the phase space) of what happens
at other times, stochastic quantum mechanics allows for a large number of alterna-
tive descriptions, and the process of refinement is often a helpful way of selecting
useful and interesting sample spaces from among them.
8.7 Unitary histories
Thus far we have discussed quantum histories without any reference to the dynam-
ical laws of quantum mechanics. The dynamics of histories is not a trivial matter,
and is the subject of the next two chapters. However, at this point it is convenient to
introduce the notion of a unitary history. The simplest example of such a history is
the sequence of kets |ψ
t
1
, |ψ
t
2
, ... |ψ
t
f
, where |ψ
t
is a solution of Schr
¨
odinger’s
equation, Sec. 7.3, or, to be more precise, the corresponding sequence of projectors
[ψ
t
1
], [ψ
t
2
],.... The general definition is that a history of the form (8.6) is unitary
provided
F
j
= T(t
j
, t
1
)F
1
T(t
1
, t
j
) (8.38)
is satisfied for j = 1, 2,... f . That is to say, all the projectors in the history
are generated from F
1
by means of the unitary time development operators intro-
duced in Sec. 7.3, see (7.44). In fact, F
1
does not play a distinguished role in this
120 Stochastic histories
definition and could be replaced by F
k
for any k, because for a set of projectors
given by (8.38), T(t
j
, t
k
)F
k
T(t
k
, t
j
) is equal to F
j
whatever the value of k.
One can also define unitary families of histories. We shall limit ourselves to the
case of a product of sample spaces, in the notation of Sec. 8.5, and assume that for
each time t
j
there is a decomposition of the identity of the form
I
j
=
a
P
a
j
. (8.39)
The corresponding family is unitary if for each choice of a these projectors satisfy
(8.38), that is,
P
a
j
= T(t
j
, t
1
)P
a
1
T(t
1
, t
j
) (8.40)
for every j. In the simplest (interesting) family of this type each decomposition of
the identity contains only two projectors; for example, [ψ
t
1
]andI − [ψ
t
1
]. Notice
that while a unitary family will contain unitary histories, such as
P
1
1
( P
1
2
( P
1
3
(···P
1
f
, (8.41)
it will also contain other histories, such as
P
1
1
( P
2
2
( P
1
3
( ···P
1
f
, (8.42)
which are not unitary. We will have more to say about unitary histories and families
of histories in Secs. 9.3, 9.6, and 10.3.
9
The Born rule
9.1 Classical random walk
The previous chapter showed how to construct sample spaces of histories for both
classical and quantum systems. Now we shall see how to use dynamical laws in
order to assign probabilities to these histories. It is useful to begin with a classical
random walk of a particle in one dimension, as it provides a helpful guide for
quantum systems, which are discussed beginning in Sec. 9.3, as well as in the next
chapter. The sample space of random walks, Sec. 8.2, consists of all sequences of
the form
s = (s
0
, s
1
, s
2
,...s
f
), (9.1)
where s
j
, an integer in the range
−M
a
≤ s
j
≤ M
b
, (9.2)
is the position of the particle or random walker at time t = j.
We shall assume that the dynamical law for the particle’s motion is that when
the time changes from t to t +1, the particle can take one step to the left, from s to
s − 1, with probability p, remain where it is with probability q, or take one step to
the right, from s to s + 1, with probability r, where
p + q +r = 1. (9.3)
The probability for hops in which s changes by 2 or more is 0. The endpoints of
the interval (9.1) are thought of as connected by a periodic boundary condition, so
that M
b
is one step to the left of −M
a
, which in turn is one step to the right of M
b
.
The dynamical law can be used to generate a probability distribution on the sample
space of histories in the following way. We begin by assigning to each history a
121
122 The Born rule
weight
W(s) =
n
j=1
w(s
j
− s
j−1
), (9.4)
where the hopping probabilities
w(−1) = p,w(0) = q,w(+1) = r (9.5)
were introduced earlier, and w(s) = 0 for |s|≥2.
The weights by themselves do not determine a probability. Instead, they must
be combined with other information, such as the starting point of the particle at
t = 0, or a probability distribution for this starting point, or perhaps information
about where the particle is located at some later time(s). This information is not
contained in the dynamical laws themselves, so we shall refer to it as contingent
information or initial data.The“initial” in initial data refers to the beginning of
an argument or calculation, and not necessarily to the earliest time in the random
walk. The single contingent piece of information “s = 3att = 2” can be the initial
datum used to generate a probability distribution on the space of all histories of the
form (9.2). Contingent information is also needed for deterministic processes. The
orbit of the planet Mars can be calculated using the laws of classical mechanics, but
to get the calculation started one needs to provide its position and velocity at some
particular time. These data are contingent in the sense that they are not determined
by the laws of mechanics, but must be obtained from observations. Once they are
given, the position of Mars can be calculated at earlier as well as later times.
Contingent information in the case of a random walk is often expressed as a
probability distribution p
0
(s
0
) on the coarse sample space of positions at t = 0. (If
the particle starts at a definite location, the distribution p
0
assigns the value 1 to
this position and 0 to all others.) The probability distribution on the refined sample
space of histories is then determined by a refinement rule that says, in essence,
that for each s
0
, the probability p
0
(s
0
) is to be divided up among all the different
histories which start at this point at t = 0, with history s assigned a fraction of
p
0
(s
0
) proportional to its weight W(s). One could also use a refinement rule if the
contingent data were in the form of a position or a probability distribution at some
later time, say t = 2ort = f , or if positions were given at two or more different
times. Things are more complicated when probability distributions are specified at
two or more times.
In order to turn the refinement rule for a probability distribution at t = 0 into a
formula, let J(s
0
) be the set of all histories which begin at s
0
, and
N(s
0
) :=
s∈J(s
0
)
W(s) (9.6)
9.1 Classical random walk 123
the sum of their weights. The probability of a particular history is then given by
the formula
Pr(s) = p
0
(s
0
)W(s)/N(s
0
). (9.7)
These probabilities sum to 1 because the initial probabilities p
0
(s
0
) sumto1,and
because the weights have been suitably normalized by dividing by the normaliza-
tion factor N(s
0
). In fact, for the weights defined by (9.4) and (9.5) using hopping
probabilities which satisfy (9.3), it is not hard to show that the sum in (9.6) is
equal to 1, so that in this particular case the normalization can be omitted from
(9.7). However, it is sometimes convenient to work with weights which are not
normalized, and then the factor of 1/N(s
0
) is needed.
Suppose the particle starts at s
0
= 2, so that p
0
(2) = 1. Then the histories (2, 1),
(2, 2), (2, 3), and (2, 4), which are compound histories for f ≥ 2, have probabili-
ties p, q, r, and 0, respectively. Likewise, the histories (2, 2, 2), (2, 2, 3), (2, 3, 4),
(2, 4, 3) have probabilities of q
2
, qr, r
2
, and 0. Any history in which the parti-
cle hops by a distance of 2 or more in a single time step has zero probability, that
is, it is impossible. One could reduce the size of the sample space by eliminating
impossible histories, but in practice it is more convenient to use the larger sample
space.
As another example, suppose that p
0
(0) = p
0
(1) = p
0
(2) = 1/3. What is the
probability that s
1
= 2 at time t = 1? Think of s
1
= 2 as a compound history given
by the collection of all histories which pass through s = 2 when t = 1, so that its
probability is the sum of probabilities of the histories in this collection. Clearly
histories with zero probability can be ignored, and this leaves only three two-time
histories: (1, 2), (2, 2), and (3, 2). In the case f = 1, formula (9.7) assigns them
probabilities r/3, q/3, and 0, so the answer to the question is (q + r)/3. This
answer is also correct for f ≥ 2, but then it is not quite so obvious. The reader
may find it a useful exercise to work out the case f = 2, in which there are nine
histories of nonzero weight passing through s = 2att = 1.
Once probabilities have been assigned on the sample space, one can answer
questions such as: “What is the probability that the particle was at s = 2 at time
t = 3, given that it arrived at s = 4 at time t = 5?” by means of conditional
probabilities:
Pr(s
3
= 2 | s
5
= 4) = Pr
(s
3
= 2) ∧ (s
5
= 4)
/ Pr(s
5
= 4). (9.8)
Here the event (s
3
= 2) ∧ (s
5
= 4) is the compound history consisting of all el-
ementary histories which pass through s = 2 at time t = 3 and s = 4 at time
t = 5. Such conditional probabilities depend, in general, both on the initial data
and the weights. However, if a value of s
0
is one of the conditions, then the condi-
tional probability does not depend upon p
0
(s
0
) (assuming p
0
(s
0
)>0, so that the