A histories approach to quantum mechanics

Stanley Gudder

Department of Mathematics and Computer Science, University of Denver,
Denver, Colorado 80208

~Received 8 June 1998; accepted for publication 14 July 1998!

A histories approach to quantum mechanics is formulated without the consistency
requirement. Plausible, physically motivated axioms for a history structure are pre-
sented. In this structure, the consistency requirement is unnecessary because the
joint sequential distributions are already probability measures. It is shown that the
logic of a history structure is a temporal effect algebra and conversely, any tempo-
ral effect algebra is isomorphic to the logic of a history structure. When this general
framework is specialized to a Hilbert space context, it reduces to a previously
studied formalism. © 1998 American Institute of Physics.
@S0022-2488~98!04411-9#

I. INTRODUCTION

The consistent histories approach to quantum mechanics has been extensively studied in

recent years.

1–8

In this article we shall delete the adjective ‘‘consistent’’ and refer to our frame-

work as a histories approach to quantum mechanics. This is because we believe that consistent sets
of histories are too restrictive and rarely occur. Even when they do occur, we question their
usefulness for describing quantum situations. However, in the absence of the consistency require-
ment, this approach may provide an important method for formulating and interpreting quantum
mechanics. Besides their emphasis on consistency and decoherence, some of these previous works
have studied temporal and logical structures in the histories framework.

1,4,5,7,8

Most recently this

approach has been extended to include generalized observables given by POV measures and the
generalized logics of effect algebras.

7,8

The goal of the present article is to provide a simplification

and unification of these previous works without the consistency requirement. Moreover, we argue
that consistent sets of histories are unnecessary in a certain sense. Unlike previous works in which
the ‘‘logic’’ of a history structure is essentially postulated, we shall derive the ‘‘logic’’ from more
basic principles.

In Sec. II we first present some plausible and physically motivated axioms for a history

structure. Although the axioms are completely different, they are motivated by Mackey’s quantum
logic approach.

9

The general framework and two of the axioms were first introduced by Medina.

10

The existence of sequential joint distributions and distributions of observables follow from the
axioms. Conditional probabilities and conditional sequential joint distributions are then defined.
Various manifestations of quantum interference are discussed.

The fundamentals of

s

-effect algebras and their associated states and observables are re-

viewed in Sec. III. A temporal effect algebra is a

s

-effect algebra that possesses a temporal

product that satisfies certain conditions. The basic properties of temporal effect algebras and their
automorphism groups are discussed.

Section IV derives the quantum logic associated with a history structure H. Histories that

occur with the same probability for every state of the system form an equivalence class that we
call a history effect and the set of history effects E is called the logic of H. It is shown that E has
the algebraic structure of a temporal effect algebra. Conversely, it is shown that any temporal
effect algebra is isomorphic to the logic of a history structure. We also prove that states and
observables in a history structure H are represented by corresponding states and observables on
the logic of H. Conditional probabilities and quantum stochastic processes on a temporal effect
algebra are discussed.

Section V considers Hilbert space history structures. If H is a complex Hilbert space and

E (H) is the set of Hilbert space effects on H, then a natural temporal product can be defined on
E (H) so that E (H) becomes a temporal effect algebra. We then carry over the concepts developed

JOURNAL OF MATHEMATICAL PHYSICS VOLUME 39, NUMBER 11 NOVEMBER 1998

5772

0022-2488/98/39(11)/5772/17/$15.00 © 1998 American Institute of Physics

Copyright ©2001. All Rights Reserved.

in Sec. IV and show that they reduce to a familiar form in E(H). Decoherence functions are
briefly discussed and some observations about them are given.

II. HISTORY STRUCTURES

Denote the

s

-algebra of Borel subsets of

R

n

by B(

R

n

). A countable Borel partition of

R

n

is

a sequence A

i

PB(R

n

) such that A

i

ùA

j

5B for iÞ j and øA

i

5R

n

. We denote the set of count-

able Borel partitions of

R

n

by P (

R

n

). If N

PP (R

n

), then the

s

-algebra generated by N is

denoted by A

~N !. Notice that A~N ! consists of all unions of sets in N and that

A(N )

#B(R

n

). It is easy to check that if N

i

PP (R), i51,...,n, then

M

5

$

A

1

3¯3A

n

: A

i

PN

i

,i

51,...,n

%

PP ~R

n

!

and we use the notation A(M)

5A(N

1

,...,N

n

). We denote the set of probability measures on

(

R

n

,A(N )), N

PP (R

n

), by M

1

1

(

R

n

,A(N )) and use the notation

M

1

1

5ø

$

M

1

1

~R

n

,A

~N !!:nPN,N PP ~R

n

!

%

.

Let O be a set of observables for a physical system. For X

PO, APB(R), we may view ~X,A! as

the event for which X has a value in the set A. We do not assume that this event is sharp. It may
be unsharp due to an imprecision in a measurement apparatus or in the case of a closed system in
which no measurement is performed, the unsharpness may be due to an intrinsic property of the
observable X. If N

PP (R), we may view (X,N ) as a set of possible alternative events that can

occur. One interpretation is that (X,N ) represents an apparatus for measuring X and the discrete
alternatives result from the dial readings of the apparatus. In the absence of a measurement, we
may interpret (X,N ) as a course-grained set of alternative events.

For X

i

PO , A

i

PB(R), i51,...,n, we call h5((X

1

,A

1

),...,(X

n

,A

n

)) a history or temporal

event and interpret h as an ordered temporal sequence in which X

i

has a value in A

i

at time

t

i

where

t

1

,

t

2

,¯,

t

n

. Alternatively, no specific times may be involved and we regard h as a

sequential event in which (X

1

,A

1

) occurs first, (X

2

,A

2

) occurs second,..., and (X

n

,A

n

) occurs

last. Notice that the A

i

could be singleton sets A

i

5

$

a

i

%

in which case ((X

1

,

$

a

1

%

),...,(X

n

,

$

a

n

%

))

would be a ‘‘path’’ in which X

i

has value a

i

, i

51,...,n. We denote the set of all histories of length

n by H

n

5(O 3B(R))

n

and the set of all histories by H

5ø

$

H

n

:n

PN

%

.

A history set is an element of H

ˆ

n

5(O3P (R))

n

for some n

PN and an sPH

ˆ

n

has the form

s

5((X

1

,N

1

),...,(X

n

,N )) where X

i

PO , N

i

PP (R), i51,...,n. We interpret s as a temporal

sequence with partitions N

i

in the temporal order

t

1

,

t

2

,¯,

t

n

or as ordered successive

measurements of observables X

i

with partitions N

i

, i

51...,n. We use the notation A

s

5A(N

1

,...,N

n

) and denote the collection of all history sets by H

ˆ 5ø

$

H

ˆ

n

:n

PN

%

. If h

5((X

1

,A

1

),...,(X

n

,A

n

)) and s

5((X

1

,N

1

),...,(X

n

,N

n

)) with A

i

PN

i

, i

51,...,n, we write h

Ps and say that the history h is an element of the history set s. Our history corresponds to a
history filter or homogeneous history and our history set corresponds to a complete set of histories
in the literature. We call (X,A)

PH

1

a one-time event and (X,N )

PH

ˆ

1

a one-time history set. If

s

5((X

1

,N

1

),...,(X

n

,N

n

)), t

5((Y

1

,M

1

),...,(Y

n

,M

n

)) are history sets, we define their se-

quential product by

st

5~~X

1

,N

1

!,...,~X

n

,N

n

!,~Y

1

,M

1

!,...,~Y

n

,M

n

!!.

Of course, st

PH

ˆ . Notice that H

ˆ becomes a semigroup under this product and by convention we

assume that

BPH

ˆ and that sB5Bs5s.

If the system is in a state

a

and h

5((X

1

,A

1

),...,(X

n

,A

n

))

PH with hPs, sPH

ˆ , then the

probability that h occurs within s is denoted by P

a,s

(A

1

3¯3A

n

). We now propose an axiomatic

framework for a statistical theory of histories. All of the axioms will concern properties of the
probabilities P

a,s

.

Let S be the set of states and O the set of observables for a physical system. A history

structure is a triple (S ,O , P) where P:S

3H

ˆ →M

1

1

, (

a

,s)

°P

a,s

and P

a,s

PM

1

1

(

R

n

,A

s

) when

s

PH

ˆ

n

. We postulate that (S ,O , P) satisfies the following axioms.

5773

J. Math. Phys., Vol. 39, No. 11, November 1998 Stanley Gudder

Copyright ©2001. All Rights Reserved.

~H1! If s

i

PHˆ

1

and P

a,s

i

(A

i

)

<P

a,s

i

11

(A

i

11

) for every

a

PS , then there exists an sPHˆ

1

and

B

i

PA

s

with B

i

#B

i

11

such that P

a,s

i

(A

i

)

5P

a,s

(B

i

), i

51,2,..., for every

a

PS .

~H2! Let s

1

,s

2

,t

1

,t

2

PH

ˆ

1

. If P

a,s

1

(A

1

)

5P

a,s

2

(A

2

) and P

a,t

1

(B

1

)

5P

a,t

2

(B

2

) for every

a

PS , then

P

a,s

1

t

1

~A

1

3B

1

!5P

a,s

2

t

2

~A

2

3B

2

!

for every

a

PS .

~H3! If s5((X

1

,N

1

),...,(X

n

,N

n

))

PHˆ and A

i

PN

i

i

51,...,n, then there exists a t5(X,N )

PHˆ

1

and an A

PN such that

P

a,t

~A!5P

a,s

~A

1

3¯3A

n

!

for every

a

PS .

~H4! If sPHˆ, t5(X,N )PHˆ

1

, u

5(X,M)PH

ˆ

1

, A

PA

s

, B

PA

t

ùA

u

, then P

a,st

(A

3B)

5P

a,su

(A

3B) for every

a

PS . Moreover, P

a,st

(A

3R)5P

a,s

(A) for every

a

PS .

~H5! If s5((X

1

,N

1

),...,(X

n

,N

n

))

PH

ˆ , t5((X

1

,M

1

),...,(X

n

,M

n

))

PH

ˆ

and A

i

PN

i

ùM

i

, then

P

a,s

~A

1

3¯3A

n

!5P

a,t

~A

1

3¯3A

n

!

for every

a

PS .

If s

5((X

1

,N

1

),...,(X

n

,N

n

))

PH

ˆ and

a

PS , we call P

a,s

PM

1

1

(

R

n

,A

s

) the sequential

joint distribution of X

1

,...,X

n

with partitions N

1

,...,N

n

in the state

a

. The main reason for

requiring a consistent set of histories in the usual approach is to ensure that P

a,s

is indeed a

probability measure. In the present approach this is not necessary because we are already guaran-
teed that P

a,s

is a probability measure. This may appear to be circumventing a problem by decree.

However, the point here is that a histories approach with proper statistical properties can be
developed without the need for decoherence functions and a consistency requirement.

We now give physical interpretations for the history structure axioms. Axiom

~H1! says that

an increasing sequence of one-time events can be described by a single observable. In this sense,
one-time events in an increasing sequence are compatible. Axiom

~H2! says that if pairs of

one-time events have the same probabilities of occurrence for every state, then so do the corre-
sponding two-time histories. Our interpretation of

~H3! is that any history can be described by a

one-time event. This is in accordance with the following statement:

7

‘‘All meaningful propositions

about a physical system specify that the value of some observable lies in some set A

PB(R).’’

The first part of

~H4! means that the partition of the last measurement is irrelevant and the second

part means that the last measurement does not affect the previous ones. This axiom as well as parts
of this general framework were given in Ref. 10. Axiom

~H5! says that the probability of a history

does not depend on the partitions and hence is unambiguous. However, if s and t are as in

~H5!

and A

PA

s

ùA

t

, then in general P

a,s

(A)

ÞP

a,t

(A). This is an unavoidable manifestation of

quantum interference and we shall mention other such manifestations in the sequel.

~Section V

discusses examples in Hilbert space.

!

It

follows

from

~H4! by letting s5B that if t5(X,N ), u5(X,M) and B

PA(N )ùA(M), then P

a,t

(B)

5P

a,u

(B) for every

a

PS . Hence, the probability that X has a

value in B is independent of the partition that generates B. The next lemma is stated without proof
in Ref. 10.

Lemma 2.1: If

a

PS , XPO , then there exists a unique probability measure P

a,X

on

~R,B~R!! such that P

a,X

(A)

5P

a,s

(A) for every s

5(X,N ) and APA(N ).

Proof: For A

PB(R), define P

a,X

(A)

5P

a,s

(A), where s

5(X,N ) with APA(N ). Our

previous observation shows that P

a,X

is well defined. Then P

a,X

(A)

>0 for every APB(R) and

P

a,X

(

R)

51. If A

i

PB(R) are mutually disjoint, i51,2,..., let N PP (R) contain each A

i

, i

51,2,... . If s5(X,N ), then

P

a,X

~øA

i

!5P

a,s

~øA

i

!5

(

P

a,s

~A

i

!5

(

P

a,X

~A

i

!.

5774 J. Math. Phys., Vol. 39, No. 11, November 1998 Stanley Gudder

Copyright ©2001. All Rights Reserved.

Hence, P

a,X

is a probability measure on

~R,B~R!! and it is clear that P

a,X

is unique.

h

We call P

a,X

the distribution of X in the state

a

. The proof of the following result is similar

to that of Lemma 2.1.

Corollary 2.2: If

a

PS , sPH

ˆ , APA

s

, X

PO, then there exists a unique probability mea-

sure P

a,s,X

such that P

a,s,X

(A

3B)5P

a,s,t

(A

3B) for every t5(X,N ) and BPA(N ).

The next lemma follows directly from

~H1!.

Lemma 2.3: If P

a,X

i

(A

i

)

<P

a,X

i

11

(A

i

11

) for every

a

PS , then there exists an XPO and

B

i

PB(R) with B

i

#B

i

11

such that P

a,X

(B

i

)

5P

a,X

i

(A

i

) for every

a

PS , i51,2,... .

By iterating

~H4! we have the following result:

10

Lemma 2.4: If

a

PS , sP((X

1

,N

1

),...,(X

n

,N ))

PHˆ and APA(N

1

), then P

a,s

(A

3R

n

21

)

5P

a,X

1

(A).

In general, the other marginal distributions are not equal to the one-variable distributions. That

is, in general for k

.1, s5((X

1

,N

1

),...,(X

n

,N

n

)) and A

PA(N

k

),

P

a,s

~R

k

21

3A3R

n

2k

!ÞP

a,X

k

~A!.

This is another manifestation of quantum interference.

If s,t

PHˆ, APA

s

, B

PA

t

with P

a,s

(A)

Þ0, we defined the conditional probability of B

given A for st in the state

a

by

P

a,tus

~BuA!5

P

a,st

~A3B!

P

a,s

~A!

.

It is clear that if t

PH

ˆ

m

, then P

a,tus

(

•uA) is a probability measure on (R

m

,A

t

) which we call a

conditional sequential joint distribution. The next lemma shows that these distributions have the
fundamental properties possessed by sequential joint distributions.

Lemma 2.5: For every

a

PS , sPH

ˆ , P

a,•us

(

•uA) satisfies ~H4! and ~H5!.

Proof: Applying

~H4! we have

P

a,tuus

~B3CuA!5

P

a,stu

~A3B3C!

P

a,s

~A!

5

P

a,stv

~A3B3C!

P

a,s

~A!

5P

a,tvus

~B3CuA!.

Thus, P

a,•us

(

•uA) satisfies the first part of ~H4!. For the second part of ~H4! we have

P

a,tuus

~B3RuA!5

P

a,stu

~A3B3R!

P

a,s

~A!

5

P

a,st

~A3B!

P

a,s

~A!

5P

a,tus

~BuA!.

The proof of

~H5! is similar. h

The next result which follows from Corollary 2.2 shows that observables have conditional

distributions on

~R,B~R!!.

Corollary 2.6: If

a

PS , XPO and P

a,s

(A)

Þ0, then there exists a unique probability mea-

sure P

a,X

(

•uA) on ~R,B~R!! such that P

a,X

(B

uA)

5P

a,tus

(B

uA) for every t

5(X,N ) and B

PA(N ).

Another manifestation of quantum interference is the fact that Bayes’ formula need not hold,

P

a,tus

~BuA!P

a,s

~A!ÞP

a,sut

~AuB!P

a,t

~B!.

This is because the left-hand side is P

a,st

(A

3B) and the right-hand side is P

a,ts

(B

3A) so the

order of the measurements or the order of the occurrences is changed. Also the other Bayes’
formula need not hold,

(

i

51

r

P

a,s

~A

i

!P

a,tus

~BuA

i

!ÞP

a,t

~B!,

where

$

A

1

,...,A

r

%

PP (R

n

), s

PHˆ

n

. This is because the left-hand side is

5775

J. Math. Phys., Vol. 39, No. 11, November 1998 Stanley Gudder

Copyright ©2001. All Rights Reserved.

(

i

51

r

P

a,st

~A

i

3B!5P

a,st

~øA

i

3B!5P

a,st

~R

n

3B!

and P

a,st

(

R

n

3B)ÞP

a,t

(B) in general. The last inequality comes from the fact that past measure-

ments can affect future ones.

For h

5((X

1

,A

1

),...,(X

n

,A

n

))

PH and

a

PS , we define P

a

(h)

5P

a,s

(A

1

3¯3A

n

) for

any s

PH

ˆ with hPs and ~H5! ensures that P

a

(h) is well defined. An automorphism on O is a

bijection

c

: O

→O that satisfies

P

a

~~

c

X,A

!!5P

a

~~

c

Y ,B

!,~

c

Z,C

!!

and

P

a

~~

c

21

X,A

!!5P

a

~~

c

21

Y ,B

!,~

c

21

Z,C

!!

for all

a

PS whenever P

a

((X,A))

5P

a

((Y ,B),(Z,C)) for all

a

PS . Thus, if a two-time history

is described by a one-time event, then

c

and

c

21

preserve this relationship. Denoting the set of

automorphisms on O by aut

~O!, it is clear that aut~O ! is a group under composition. A dynamical

group on O is a group homomorphism

t

°

c

(

t

) from the additive group

R into aut

~O!. We thus

have

c

(

t

1

1

t

2

)

5

c

(

t

1

)

+

c

(

t

2

) for all

t

1

,

t

2

PR and

c

(0)

5I, where I is the identity automor-

phism. We interpret

c

(

t

)X as the evolution of the observable X. If X

1

,...,X

n

are observables at

time

t

50 and

t

1

,...,

t

n

, we interpret

~~

c

~

t

1

!X

1

,A

1

!,...,~

c

~

t

n

!X

n

,A

n

!!

as a history in which X

1

has a value in A

1

at time

t

1

,...,X

n

has a value in A

n

at time

t

n

.

III. TEMPORAL EFFECT ALGEBRAS

An effect algebra

8,11–16

is an algebraic system (E,0,1,

%

) where 0,1

PE and

%

is a partial

binary operation on E that satisfies

~E1! if a

%

b is defined, then b

%

a is defined and b

%

a

5a

%

b;

~E2! if a

%

b and (a

%

b)

%

c are defined, then b

%

c and a

%

(b

%

c) are defined and a

%

(b

%

c)

5(a

%

b)

%

c;

~E3! for every aPE, there exists a unique a

8

PE such that a

%

a

8

is defined and a

%

a

8

51;

~E4! if a

%

1 is defined, then a

50.

We call a

%

b an orthogonal sum. It follows from

~E2! that existing orthogonal sums do not

require parentheses and we write a

%

b

%

c for (a

%

b)

%

c. More generally, we write a

1

%

¯

%

a

n

if

this orthogonal sum is defined. We define a

<b if there exists a cPE such that a

%

c

5b. It can be

shown

13

that (E,0,1,

<) is a bounded poset and a

%

b is defined if and only if a

<b

8

. If a

<b

8

, we

write a

'b. If a<b, then the unique c that satisfies a

%

c

5b is denoted by c5b

*

a. If the least

upper bound

∨

a

i

exists whenever a

1

<a

2

<¯ , then E is called a

s

-effect algebra.

8

Suppose b

i

5a

i

%

¯

%

a

i

is defined for all i

PN. If

∨

b

i

exists then we say that

%

a

i

exists and we define

%

a

i

to be

∨

b

i

.

Lemma 3.1: An effect algebra E is a

s

-effect algebra if and only if

%

a

i

exists whenever a

1

%

¯

%

a

i

is defined for all i

PN.

Proof: If E is a

s

-effect algebra and b

i

5a

1

%

¯

%

a

i

is defined for all i

PN, then b

1

<b

2

<..., so

∨

b

i

exists. Conversely, suppose

%

a

i

exists whenever a

1

%

¯

%

a

i

is defined for all i

PN and let b

1

<b

2

<¯ . Defining a

1

5b

1

, a

i

5b

i*

b

i

21

, i

52,3,..., it easily follows that a

1

%

¯

%

a

i

5b

i

exists for all i

PN. Hence,

∨

b

i

5

%

a

i

, exists so E is a

s

-effect algebra.

h

Let E and F be

s

-effect algebras. A map

f

: E

→F is additive if a'b implies

f

(a)

'

f

(b) and

f

(a

%

b)

5

f

(a)

%

(b). If

f

is additive, then

f

is monotone in the sense that a

<b implies

f

(a)

<

f

(b). Indeed, if a

<b then b5a

%

c for some c

PE so that

f

~a!<

f

~a!

%

f

~c!5

f

~a

%

c

!5

f

~b!.

5776 J. Math. Phys., Vol. 39, No. 11, November 1998 Stanley Gudder

Copyright ©2001. All Rights Reserved.

An additive map

f

: E

→F is

s

-additive if a

1

<a

2

<¯ implies that

f

(

∨

a

i

)

5

∨

f

(a

i

). The proof

of the next result is similar to that of Lemma 3.1.

Lemma 3.2: A map

f

: E

→F is

s

-additive if and only if

f

(

%

a

i

)

5

%

f

(a

i

) whenever

%

a

i

exists.

A

s

-additive map

f

: E

→F is a

s

-morphism if

f

(1)

51. It is clear that a

s

-morphism satisfies

f

(a

8

)

5

f

(a)

8

. If

f

: E

→F is a bijective

s

-morphism and

f

21

is a

s

-morphism, then

f

is a

s

-isomorphism.

An important example of a

s

-effect algebra is the unit interval

@0,1##R. For a,bP@0,1# we

say that a

%

b is defined if a

1b<1 and in this case a

%

b

5a1b. We then have a

8

512a, 0 is the

zero element and 1 is the one element. A

s

-morphism from a

s

-effect algebra E into

@0,1# is called

a state on E. We denote the set of states on E by S (E). We say that S

#S (E) is order deter-

mining if

a

(a)

<

a

(b) for all

a

PS implies that a<b.

Another example of a

s

-effect algebra is a measurable space

~V,A!. For A,BPA we say that

A

%

B is defined if A

ùB5B and in this case A

%

B

5AøB. We then have A

8

equal to the

complement of A and 0

5B, 15V. A

s

-morphism from

~V,A! into a

s

-effect algebra E is called

an observable on E with value space

~V,A!. We denote the set of such observables by O

A

(E).

Notice that S

~V,A! is precisely the set of probability measures on ~V,A!. If XPO

A

(E) and

a

PS (E), then

a

+XPS (V,A) is called the distribution of X in the state

a

.

The elements of a

s

-effect algebra are called effects. If

a

PS (E), then

a

(a) is the probability

that a occurs when the system is in the state

a

. For X

PO

A

(E), X(A) is the effect that X has a

value in A and

a

+X(A) is the probability that X(A) occurs in the state

a

. We interpret a

%

b as a

statistical sum in the sense that a

%

b is the effect that satisfies

a

(a

%

b)

5

a

(a)

1

a

(b) for every

state

a

.

A temporal product on a

s

-effect algebra E is a binary operation (a,b)

°a+b on E such that

b

°a+b is

s

-additive and 0

+a5a, a+15a for every aPE. Notice that a+050 for every aPE

because

a

5a+15a+~0

%

1

!5a+0

%

a

+15a+0

%

a.

Hence, by cancellation

13

a

+050. We interpret a+b as the effect in which a first occurs and then

b occurs. A natural temporal product on the

s

-effect algebra

~V,A! is given by A+B5AùB. The

next lemma shows that there is only one temporal product on

@0,1#.

Lemma 3.3: The unique temporal product on

@0,1# has the form a+b5ab.

Proof: It is clear that a

+b5ab is a temporal product on @0,1#. Conversely, suppose a+b is a

temporal product on

@0,1#. Let aP@0,1# with aÞ0 and define

f

a

:

@0,1#→@0,1# by

f

a

(b)

5(a

+b)/a. Then

f

a

PS (@0,1#) and it follows

14

that

f

a

(b)

5b for all bP@0,1#. Hence, a+b5ab. h

In general, a temporal product is not associative. When we write a

1

+a

2

+¯+a

n

we mean that

the operations are performed from right to left. That is, first a

n

21

+a

n

is performed, then a

n

22

+(a

n

21

+a

n

) is performed,..., and finally a

1

+(a

2

+¯+a

n

) is performed. For example, a

+b+c means

a

+(b+c).

Lemma 3.4: Let

+ be a temporal product on E. ~i! a+b<a for all a,bPE. ~ii! a+b

8

5a

*

a

+b

for all a,b

PE. ~iii! If a<b then c+a<c+b for all cPE. ~iv! If a<b, then c+(b

*

a)

5c+b

*

c

+a

for all c

PE.

Proof:

~i! For a,bPE we have

a

5a+15a+~b

%

b

8

!5a+b

%

a

+b

8

>a+b.

~ii! This follows from the proof of ~i!. ~iii! The function a°c+a is additive so it is monotone. ~iv!
Since b

5a

%

(b

*

a) we have

c

+a

%

c

+~b

*

a

!5c+~a

%

~b

*

a

!!5c+b.

h

A temporal effect algebra is a

s

-effect algebra that possesses an order determining set of

states and a temporal product. For the remainder of this section, E will be a temporal effect algebra
with temporal product

+. For

a

PS (E) and a,bPE with

a

(a)

Þ0, the conditional probability of

b given a in the state

a

is defined as

a

(b

ua)

5

a

(a

+b)/

a

(a). Notice that

a

(

•ua)PS (E) and we

denote this state by

a

ua. Thus, (

a

ua)(b)

5

a

(a

+b)/

a

(a). By employing the temporal product

5777

J. Math. Phys., Vol. 39, No. 11, November 1998 Stanley Gudder

Copyright ©2001. All Rights Reserved.

A

+B5AùB on ~V,A! we see that this definition of conditional probability reduces to the usual

concept. Also, it is easy to check that

~V,A! and @0,1# are temporal effect algebras.

Temporal effect algebras also give a convenient framework for discussing dynamics. An

automorphism on E is a

s

-isomorphism

f

: E

→E such that

f

(a

+b)5

f

(a)

+

f

(b) for all a,b

PE. Notice that the set of automorphisms aut(E) on E forms a group under composition. Indeed,
it is clear that if

f

,

c

Paut(E), then

fc

Paut(E). Moreover, if

f

Paut(E), then

f

21

is a

s

-

isomorphism and

f

@

f

21

~a!+

f

21

~b!#5a+b

so that

f

21

~a!+

f

21

~b!5

f

21

~a+b!.

Hence,

f

21

Paut(E). If

f

1

,...,

f

n

Paut(E), a

1

,...,a

n

PE, we use the notation

f

1

+a

1

+

f

2

+a

2

+¯+

f

n

+a

n

to mean that each

f

i

operates on the expression to its right. For example,

f

1

+a

1

+

f

2

+a

2

5

f

1

@a

1

+~

f

2

~a

2

!!#.

A dynamical group on E is a one-parameter group of automorphisms on E. That is, a dynamical
group on E is a group homomorphism

t

°

f

(

t

) from the additive group

R into aut(E). We thus

have

f

(

t

1

1

t

2

)

5

f

(

t

2

)

+

f

(

t

2

) for all

t

1

,

t

2

PR and

f

(0)

5I where I is the identity automor-

phism. It follows that

f

(

2

t

)

5

f

(

t

)

21

for all

t

PR. If aPE is an effect at time

t

50, we

interpret

f

(

t

)a as the evolution of a at time

t

. If a

1

,...,a

n

PE correspond to effects at time

t

50 and

t

1

,¯,

t

n

, we interpret

f

(

t

1

)a

1

+¯+

f

(

t

n

)a

n

as the effect in which a

1

occurs at time

t

1

,a

2

at time

t

2

,...,a

n

at time

t

n

.

Lemma 3.5: If

f

~

t

! is a dynamical group, then

f

~

t

1

!a

1

+¯+

f

~

t

n

!a

n

5

f

~

t

1

!+a

1

+

f

~

t

2

2

t

1

!+a

2

+¯+

f

~

t

n

2

t

n

21

!+a

n

.

Proof: We proceed by induction on n. The result clearly holds for n

51. For n52 we have

f

~

t

1

!a

1

+

f

~

t

2

!a

2

5

f

~

t

1

!a

1

+

f

~

t

1

!

f

~

t

2

2

t

1

!a

2

5

f

~

t

1

!@a

1

+

f

~

t

2

2

t

1

!a

2

#5

f

~

t

1

!+a

1

+

f

~

t

2

2

t

1

!+a

2

.

Now suppose the result holds for n

>1. Then

f

~

t

1

!a

1

+¯+

f

~

t

n

11

!a

n

11

5

f

~

t

1

!a

1

+

f

~

t

2

!@a

2

+

f

~

t

3

2

t

2

!+a

3

+¯+

f

~

t

n

11

2

t

n

!+a

n

11

#

5

f

~

t

1

!+a

1

+

f

~

t

2

2

t

1

!+a

2

+¯+

f

~

t

n

11

2

t

n

!a

n

11

.

The result follows by induction.

h

If

t

1

,¯,

t

n

, we can interpret Lemma 3.5 as saying that the system evolves until time

t

1

and a

1

occurs, then the system evolves from time

t

1

until time

t

2

and a

2

occurs,..., then the

system evolves from time

t

n

21

until time

t

n

and a

n

occurs.

If

f

Paut(E), we define

f

ˆ : S (E)→S (E) by

f

ˆ

a

(a)

5

a

(

f

21

a). Then

f

ˆ is a bijection on

S (E). Moreover, (

f

1

f

2

)

∧

5

f

ˆ

1

f

ˆ

2

because

~

f

1

f

2

!

∧

a

~a!5

a

~~

f

1

f

2

!

21

a

!5

a

~

f

2

21

f

1

21

a

!5

f

ˆ

2

a

~

f

1

21

a

!5

f

ˆ

1

f

ˆ

2

a

~a!.

Hence, aut(E)

∧

is a group and

f

°

f

ˆ is a group isomorphism from aut(E) onto aut(E)

∧

. If

f

~

t

!

is a dynamical group on E and

a

PS (E) is the state of the system at time

t

50, then we interpret

f

ˆ (

t

0

)

a

as the state at time

t

0

. If 0

,

t

0

,

t

1

,¯,

t

n

, by Lemma 3.5, the probability of the

temporal effect

f

(

t

1

)a

1

+¯+

f

(

t

n

)a

n

in the state

f

ˆ (

t

0

)

a

becomes

f

ˆ ~

t

0

!

a

@

f

~

t

1

!+a

1

+¯+

f

~

t

n

2

t

n

2a

!+a

n

#5

a

@

f

~

t

1

2

t

0

!+a

1

+¯+

f

~

t

n

2

t

n

21

!+a

n

#.

5778 J. Math. Phys., Vol. 39, No. 11, November 1998 Stanley Gudder

Copyright ©2001. All Rights Reserved.

We can also transfer an automorphism on E into a bijection on O (E)

5O

B(

R)

(E). An ob-

servable automorphism is a bijection

c

: O (E)

→O(E) such that (

c

X)(A)

5(

c

Y )(B)

+(

c

Z)(C)

and (

c

21

X)(A)

5(

c

21

Y )(B)

+(

c

21

Z)(C) whenever X(A)

5Y(B)+Z(C). It is clear that the set

of observable automorphisms aut

O

(E) is a group under composition. If

f

Paut(E), define

f

˜ :

O (E)

→O(E) by (

f

˜ X)(A)5

f

@X(A)#.

Theorem 3.6: The map

f

°

f

˜ is a group isomorphism from aut(E) onto aut

O

(E).

Proof: Letting

f

Paut(E), it is straightforward to show that

f

˜ : O (E)→O(E) is a bijection.

To show that

f

˜ Paut

O

(E) suppose X(A)

5Y(B)+Z(C). We then have

~

f

˜ X!~A!5

f

@X~A!#5

f

@Y~B!+Z~C!#5

f

@Y~B!#+

f

@Z~C!#5~

f

˜ Y !~B!+~

f

˜ Z!~C!.

Also, since (

f

˜ )

21

5(

f

21

)˜ we have

~~

f

˜ !

21

X

!~A!5~~

f

˜ !

21

Y

!~B!+~~

f

˜ !

21

Z

!~C!

so

f

˜ Paut

O

(E). If

f

1

,

f

2

Paut(E), then

@~

f

1

f

2

!˜ X#~A!5

f

1

f

2

@X~A!#5

f

1

~

f

˜

2

X

!~A!5~

f

˜

1

f

˜

2

X

!~A!.

Hence, (

f

1

f

2

)

;

5

f

˜

1

f

˜

2

so

f

°

f

˜ is a group homomorphism. To show that

f

°

f

˜ is injective

suppose that

f

˜

1

5

f

˜

2

and a

PE. Let l

1

,

l

2

PR with l

1

Þl

2

and define X

PO(E) by X(l

1

)

5a, X(l

2

)

5a

8

and for A

PB(R)

X

~A!5

%

$

X

~l

i

!:l

i

PA,i51,2

%

.

We then have

f

1

~a!5

f

1

@X~l

1

!#5~

f

˜

1

X

!~l

1

!5~

f

˜

2

X

!~l

1

!5

f

2

@X~l

1

!#5

f

2

~a!.

Hence,

f

1

5

f

2

so

f

°

f

˜ is injective. Finally, to show that

f

°

f

˜ is surjective, suppose

c

Paut

O

(E). We have previously shown that for a

PE there exists an XPO(E) and an A

PB(R) such that X(A)5a. Define

f

: E

→E by

f

(a)

5(

c

X)(A). Now

f

is well defined because

if Y (B)

5a we have X(A)5Y(B)+Y(R) so that

~

c

X

!~A!5~

c

Y

!~B!+~

c

Y

!~R!5~

c

Y

!~B!.

To show that

f

is injective, suppose

f

(a)

5

f

(b). As before there exists X,Y

PO (E),l

1

,

l

2

PR such that X(l

1

)

5a, X(l

2

)

5a

8

, Y (

l

1

)

5b, Y(l

2

)

5b

8

. Since

X

~

$

l

1

,

l

2

%

!515X~R!+X~R!,

we have

~

c

X

!~l

1

!

%

~

c

X

!~l

2

!5~

c

X

!~

$

l

1

,

l

2

%

!5~

c

X

!~R!+~

c

X

!~R!51,

and similarly, (

c

Y )(

l

1

)

%

(

c

Y )(

l

2

)

51. Moreover,

~

c

X

!~l

1

!5

f

~a!5

f

~b!5~

c

Y

!~l

1

!

and

~

c

X

!~l

2

!5~

c

X

!~l

1

!

8

5~

c

Y

!~l

1

!

8

5~

c

Y

!~l

2

!.

It follows that

c

X

5

c

Y and since

c

is injective, we have X

5Y. Hence, a5b so

f

is injective. To

show that

f

is surjective, suppose b

PE. Then there exist YPO (E), lPR such that Y(l)5b.

Since

c

is surjective, there exists an X

PO (E) such that

c

X

5Y. Letting a5X(l) we have

f

~a!5~

c

X

!~l!5Y~l!5b,

5779

J. Math. Phys., Vol. 39, No. 11, November 1998 Stanley Gudder

Copyright ©2001. All Rights Reserved.

so

f

is surjective. Since X(

R)

51 for XPO(E) we have

f

~1!5~

c

X

!~R!51.

Assume that a

i

PE, i51,2,..., and

%

a

i

exists. Let

l

0

,

l

1

,...

PR be distinct and define X

PO (E) by X(l

0

)

5(

%

a

i

)

8

, X(

l

i

)

5a

i

, i

51,2,..., and for APB(R),

X

~A!5

%

$

X

~l

i

!:l

i

PA

%

.

We then have

f

~

%

a

i

!5~

c

X

!~ø

$

l

i

%

!5

%

~

c

X

!~l

i

!5

%

f

~a

i

!,

and it follows from Lemma 3.2 that

f

is

s

-additive. Define

f

1

: E

→E by

f

1

(a)

5(

c

21

X)(A)

where X(A)

5a. As before,

f

1

is a well defined

s

-morphism and we have

ff

1

~a!5~

cc

21

X

!~A!5X~A!5a.

Hence,

ff

1

5I and similarly,

f

1

f

5I. It follows that

f

is a

s

-isomorphism. For a,b

PE there

exists X,Y ,Z

PO (E), A,B,CPB(R) such that X(A)5a, Y(B)5b, Z(C)5a+b. Since Z(C)

5X(A)+Y(B) we have

f

~a+b!5

f

~Z~C!!5~

c

Z

!~C!5~

c

X

!~A!+~

c

Y

!~B!5

f

~a!+

f

~b!.

Hence,

f

Paut(E) and for any XPO(E), APB(R) we have

~

c

X

!~A!5

f

@X~A!#5~

f

˜ X!~A!.

We conclude that

c

5

f

˜ so

f

°

f

˜ is surjective. h

A quantum stochastic process on E is a set of observables X

t

,

t

PR. If

f

~

t

! is a dynamical

group on E and X

PO (E) then X

t

5

f

(

t

)˜ (X) is an example of a quantum stochastic process. We

then interpret X

t

as the evolution of the observable X.

IV. HISTORY EFFECTS

This section shows that there is a close connection between a history structure and a temporal

effect algebra. In this section (S ,O , P) will denote a fixed history structure with set of histories
H. We have seen that for h

PH,

a

PS , the probability P

a

(h) is well defined. Moreover, by

~H3! there exists a one-time event ~X,A! such that

P

a

~h!5P

a,X

~A!5P

a

~~X,A!!

for every

a

PS . For h

1

,h

2

PH we define h

1

;h

2

if P

a

(h

1

)

5P

a

(h

2

) for every

a

PS . In this

case h

1

and h

2

are statistically indistinguishable. Clearly,

; is an equivalence relation on H and

we denote the equivalence class containing h by

@h#. We call the elements of E5H/;

5

$

@h#:hPH

%

history effects and we call E the logic for (S ,O , P). Notice that for every

@h#

PE there exists a one-time effect @~X,A!# such that @h#5@(X,A)#. For

a

PS , @h#PE, we define

P

a

(

@h#)5P

a

(h). We also define O

PE by O5@(X,B)# and 1PE by 15@(X,R)# for any X

PO .

Lemma 4.1:

~i! For a,bPE, if P

a

(a)

5P

a

(b) for every

a

PS , then a5b. ~ii! For a,b

PE, if P

a

(a)

1P

a

(b)

<1 for every

a

PS , then there exists a unique cPE such that P

a

(c)

5P

a

(a)

1P

a

(b) for every

a

PS .

Proof:

~i! Letting a5@h

1

#, b5@h

2

# we have P

a

(h

1

)

5P

a

(h

2

) for every

a

PS . Hence, h

1

;h

2

so a

5b. ~ii! Let a5@(X,A)#, b5@(Y,B)#. Then for every

a

PS we have

P

a,X

~A!5P

a

~a!<12P

a

~b!512P

a,Y

~B!5P

a,Y

~B

8

!.

By Lemma 2.3, there exists a Z

PO and C, DPB(R) with C#D such that P

a,Z

(C)

5P

a,X

(A)

and P

a,Z

(D)

5P

a,Y

(B

8

) for every

a

PS . Since CùD

8

5B, letting c5@(Z,CøD

8

)

# we have

5780 J. Math. Phys., Vol. 39, No. 11, November 1998 Stanley Gudder

Copyright ©2001. All Rights Reserved.

P

a

~c!5P

a,Z

~CøD

8

!5P

a,Z

~C!1P

a,Z

~D

8

!5P

a,X

~A!1P

a,Y

~B!5P

a

~a!1P

a

~b!

for every

a

PS . The uniqueness of c follows from ~i!.

h

For a, b

PE, we write a'b if P

a

(a)

1P

a

(b)

<1 for every

a

PS . If a'b we define a

%

b

5c, where c is the unique element of E given by Lemma 4.1~ii!. We interpret c as the statistical
sum of a and b in the sense that c is the unique element of E such that P

a

(c)

5P

a

(a)

1P

a

(b) for

every

a

PS . We define the temporal product of @(X,A)#,@(Y,B)#PE by

@~X,A!#+@~Y,B!#5@~~X,A!,~Y,B!!#.

The temporal product is well defined by

~H2!. For a,bPE we have a5@(X,A)#, b5@(Y,B)# for

some X,Y

PO and A,BPB(R) and we define a+b5@(X,A)#+@(Y,B)#. The next result shows that

the logic of a history structure is a temporal effect algebra.

Theorem 4.2: The algebraic system

~E,0,1,

%

,

+! is a temporal effect algebra and

$

P

a

:

a

PS

%

is an order determining set of states on E.

Proof: It is clear that if a

'b, then b'a and b

%

a

5a

%

b. Now suppose that a

'b and a

%

b

'c. Then for every

a

PS we have

P

a

~a!1P

a

~b!1P

a

~c!5P

a

~a

%

b

!1P

a

~c!<1

so b

'c. Since P

a

(a)

1P

a

(b

%

c)

<1 for every

a

PS , we have a'b

%

c. For every

a

PS we

have

P

a

~a

%

~b

%

c

!!5P

a

~a!1P

a

~b!1P

a

~c!5P

a

~~a

%

b

!

%

c

!

so by Lemma 4.1

~i!, a

%

(b

%

c)

5(a

%

b)

%

c. For a

5@(X,A)#PE, let a

8

5@(X,A

8

)

#. Then for

every

a

PS we have

P

a

~a!1P

a

~a

8

!5P

a,X

~A!1P

a,X

~A

8

!515P

a

~1!.

Hence, a

'a

8

and by Lemma 4.1

~i!, a

%

a

8

51 and a

8

is unique. If a

'1, then for every

a

PS we

have

P

a

~a!115P

a

~a!1P

a

~1!<1.

Hence, P

a

(a)

50 for every

a

PS so a50. We conclude that ~E,0,1,

%

! is an effect algebra. For

a

PS , we have P

a

(1)

51 and P

a

is additive because a

'b implies that P

a

(a

%

b)

5P

a

(a)

1P

a

(b). To show that

$

P

a

:

a

PS

%

is order determining suppose that P

a

(a)

<P

a

(b) for every

a

PS . Then P

a

(a)

1P

a

(b

8

)

<1 for every

a

PS so a'b

8

. Since

P

a

~a!1P

a

~~a

%

b

8

!

8

!5P

a

~a!112P

a

~a

%

b

8

!5P

a

~a!112P

a

~a!2P

a

~b

8

!5P

a

~b!

for every

a

PS we have a

%

(a

%

b

8

)

8

5b so a<b. To show that E is a

s

-effect algebra, suppose

that a

i

PE, i51,2,..., with a

1

<a

2

<¯ . Then a

i

5@(X

i

,A

i

)

# and since P

a

(a

i

)

<P

a

(a

i

11

) we

have P

a,X

i

(A

i

)

<P

a,X

i

11

(A

i

11

), i

51,2,..., for every

a

PS . Applying Lemma 2.3, there exists an

X

PO and B

i

PB(R) with B

i

#B

i

11

such that P

a,X

(B

i

)

5P

a,X

i

(A

i

), i

51,2,..., for every

a

PS .

Letting a

5@(X,øB

i

)

# we have

lim P

a

~a

i

!5lim P

a,X

i

~A

i

!5lim P

a,X

~B

i

!5P

a,X

~øB

i

!5P

a

~a!

for every

a

PS . Since P

a

(a

i

)

<P

a

(a) for every

a

PS , we conclude that a

i

<a, i51,2,... . If

a

i

<b, i51,2,..., then P

a

(a

i

)

<P

a

(b) for every

a

PS , i51,2,... . Hence, P

a

(a)

<P

a

(b) for

every

a

PS so a<b. Hence, a5

∨

a

i

so E is a

s

-effect algebra. Also P

a

is a state for every

a

PS because we have just shown that if a

1

<a

2

<¯ , then P

a

(

∨

a

i

)

5lim P

a

(a

i

).

To complete the proof, we must show that

+ is indeed a temporal product. Letting a

5@(X,A)#PE, then for every

a

PS we have by ~H4! that

P

a

~a+1!5P

a

~@~X,A!,~X,R!#!5P

a,s

~A3R!5P

a,X

~A!5P

a

~a!.

5781

J. Math. Phys., Vol. 39, No. 11, November 1998 Stanley Gudder

Copyright ©2001. All Rights Reserved.

Hence, a

+15a for every aPE. Moreover,

P

a

~0+a!5P

a

~@~X,B!,~X,A!#!5P

a,s

~B3A!5P

a,s

~B!50

for every

a

PS , so 0+a50. To show that + is additive, let b'c so that b<c

8

. Since P

a

(b)

<P

a

(c

8

) for every

a

PS , by Lemma 2.3 there exists an XPO and B, DPB(R) with B#D such

that P

a,X

(B)

5P

a

(b), P

a,X

(D)

5P

a

(c

8

) for every

a

PS . Then,

P

a,X

~BøD

8

!5P

a,X

~B!1P

a,X

~D

8

!5P

a

~b!1P

a

~c!5P

a

~b

%

c

!

for every

a

PS . Hence, @(X,BøD

8

)

#5b

%

c. For a

5@(Y,A)#PE, by Corollary 2.2 we have

P

a

~a+~b

%

c

!!5P

a,s,X

~@~~Y,A!,~X,BøD

8

!!#!

5P

a,s,X

~@~~Y,A!,~X,B!!#!1P

a,s,X

~@~~Y,A!,~X,D

8

!!#!

5P

a

~a+b!1P

a

~a+c!

for every

a

PS . We conclude that a+b'a+c and a+(b

%

c)

5a+b

%

a

+c. To show that + is

s

-

additive, suppose that b

1

<b

2

<... . By Lemma 3.4~iii! we have that a+b

1

<a+b

2

<... . Applying

~H1! there exists YPO and B

i

PB(R) with B

i

#B

i

11

such that P

a,Y

(B

i

)

5P

a

(b

i

), i

51,2,..., for

every

a

PS . As in the previous paragraph, we have @(Y,øB

i

)

#5

∨

b

i

. Letting a

5@(X,A)# we

have by Corollary 2.2 that

P

a

~

∨

a

+b

i

!5lim P

a

~a+b

i

!5lim P

a,s,Y

~@~~X,A!,~Y,B

i

!!#!

5P

a,s,Y

~@~~X,A!,~Y,øB

i

!!#!5P

a

~a+

∨

b

i

!

for every

a

PS . Hence,

∨

a

+b

i

5a+

∨

b

i

.

h

For X

PO, define Xˆ:B(R)→E by Xˆ(A)5@(X,A)#.

Theorem 4.3: The map X

ˆ :B(R)→E is an observable on E and the distribution P

a,X

of X

coincides with the distribution P

a

+Xˆ of Xˆ for every

a

PS .

Proof: Notice that X

ˆ (R)5@(X,R)#51. If A, BPB(R) with AùB5B, then

P

a,X

~A!1P

a,X

~B!5P

a,X

~AøB!<1

for every

a

PS . Hence, @(X,A)#'@(X,B)# and

P

a

~@~X,A!#!1P

a

~@~X,B!#!5P

a

~@~X,AøB!#!

for every

a

PS so that Xˆ(AøB)5Xˆ(A)

%

X

ˆ (B). Suppose that A

i

PB(R) with A

i

ùA

j

5B, i

Þ j. Then for nPN we have

X

ˆ

S

ø

i

51

`

A

i

D

>Xˆ

S

ø

i

51

n

A

i

D

5

%

i

51

n

X

ˆ ~A

i

!.

Suppose that a

>

%

i

51

n

X

ˆ (A

i

) for all n

PN. Then

P

a

~a!>

(

i

51

n

P

a,X

~A

i

!

for all n

PN so that

P

a

~a!>

(

i

51

`

P

a,X

~A

i

!5P

a,X

S

ø

i

51

`

A

i

D

5P

a

S

X

ˆ

S

ø

i

51

`

A

i

DD

for every

a

PS . Hence, a>Xˆ(øA

i

) so that

5782 J. Math. Phys., Vol. 39, No. 11, November 1998 Stanley Gudder

Copyright ©2001. All Rights Reserved.

X

ˆ ~øA

i

!5

∨

%

i

51

n

X

ˆ ~A

i

!5

%

i

51

`

X

ˆ ~A

i

!.

It follows from Lemma 3.2 that X

ˆ PO(E). Finally, for

a

PS , APB(R) we have

~P

a

+Xˆ!~A!5P

a

~@~X,A!#!5P

a

~~X,A!!5P

a,X

~A!.

h

It follows from Theorem 4.2 that any state

a

PS can be represented by a state P

a

PS (E).

Moreover, by Theorem 4.3, any observable X

PO can be represented by an observable Xˆ

PO(E). We can also represent history sets in H

ˆ

by

observables

on

E .

If

s

5((X

1

,N

1

),...,(X

n

,N

n

))

PHˆ and A

i

PN

i

, i

51,...,n, define

sˆ

~A

1

3¯3A

n

!5@~~X

1

,A

1

!,...,~X

n

,A

n

!!#PE.

Then as in the proof of Theorem 4.3, sˆ extends to a unique observable on E with value space
(

R

n

,A

s

). Moreover, the joint sequential distribution P

a,s

of s coincides with the distribution P

a

+sˆ

of sˆ for every

a

PS .

We have shown that the logic of a history structure is a temporal effect algebra. We now

prove the converse statement. That is, any temporal effect algebra is

s

-isomorphic to the logic of

a history structure.

Theorem 4.4: If (E,0,1,

%

,

+) is a temporal effect algebra, then there exists a history structure

(S ,O , P) such that E is

s

-isomorphic to the logic of (S ,O , P).

Proof: Let H, H

ˆ be the families of histories and history sets, respectively, for O (E). That is,

H

5ø

$

~O~E!3B~R!!

n

:n

PN

%

,

H

ˆ 5ø

$

~O~E!3P ~R!!

n

:n

PN

%

.

For s

5((X

1

,N

1

),...,(X

n

,N

n

))

PH

ˆ ,

a

PS (E) and A

i

PN

i

, i

51,...,n, define

P

a,s

~A

1

3¯3A

n

!5

a

~X

1

~A

1

!+¯+X

n

~A

n

!!.

We now show that P

a,s

extends uniquely to an element of M

1

1

(

R

n

,A

s

). Suppose that N

i

5

$

A

i

j

: j

51,2,...

%

, i

51,...,n. By the

s

-additivity of

+ we have

%

j

n

X

1

~A

1

j

1

!+¯+X

n

~A

n

j

n

!5X

1

~A

1

j

1

!+¯+X

n

21

~A

n

21

j

n

21

!+

%

j

n

X

n

~A

n

j

n

!

5X

1

~A

1

j

1

!+¯+X

n

21

~A

n

21

j

n

21

!+X

n

S

ø

j

n

A

n

j

n

D

5X

1

~A

1

j

1

!+¯+X

n

21

~A

n

21

j

n

21

!.

Proceeding by induction, we conclude that

%

j

1

,..., j

n

X

1

~A

1

j

1

!+¯+X

n

~A

n

j

n

!51.

Hence,

(

j

1

,..., j

n

P

a,s

~A

1

j

1

3¯3A

n

j

n

!51.

Now any A

PA

s

has the form

A

5ø

$

A

1

j

1

3¯3A

n

j

n

: j

1

PI

1

,..., j

n

PI

n

%

,

where I

i

#N is an index set depending on A, i51,...,n. We then define P

a,s

PM

1

1

(

R

n

,A

s

) by

5783

J. Math. Phys., Vol. 39, No. 11, November 1998 Stanley Gudder

Copyright ©2001. All Rights Reserved.

P

a,s

~A!5

(

$

P

a,s

~A

1

j

1

3¯3A

n

j

n

!: j

1

PI

1

,..., j

n

PI

n

%

.

We now show that (S (E),O (E), P) is a history structure. To verify

~H1!, suppose that s

i

5(X

i

,N

i

)

PHˆ

1

and P

a,s

i

(A

i

)

<P

a,s

i

11

(A

i

11

) for every

a

PS (S), i51,2,... . Letting a

i

5X

i

(A

i

) we have

a

~a

i

!5

a

~X

i

~A

i

!!5P

a,s

i

~A

i

!<P

a,s

i

11

~A

i

11

!5

a

~X

i

11

~A

i

11

!!5

a

~a

i

11

!

for every

a

PS (E), i51,2,... . Since S (E) is order determining, we have a

i

<a

i

11

, i

51,2,... .

Let a

5

∨

a

i

, b

1

5a

1

, b

i

5a

i

*a

i

21

, i

52,3,..., and we have

%

b

i

5a. Letting b

0

5a

8

, we have

%

i

50

`

b

i

51. Now let

$

l

i

:i

50,1,...

%

#R, define X(l

i

)

5b

i

, i

50,1,..., and for BPB(R) define

X

~B!5

%

$

X

~l

i

!:l

i

PB

%

.

Then X

PO(E) and for B

i

5

$

l

1

,...,

l

i

%

, i

51,2,..., we have B

i

#B

i

11

and

X

~B

i

!5

%

j

51

i

X

~l

j

!5

%

j

51

i

b

5a

i

.

Letting N be a partition generated by

$

l

i

%

i

51,2,..., and s5(X,N ) we have

P

a,s

i

~A

i

!5

a

~a

i

!5

a

~X~B

i

!!5P

a,s

~B

i

!

for every

a

PS (E). The verifications of ~H2!–~H5! are straightforward. Moreover, it is easy to

show that the logic of (S (E),O (E), P) is

s

-isomorphic to E.

h

It follows from Theorem 4.4 that a temporal effect algebra E generates a history structure

(S (E),O (E), P) in a natural way. For X

i

PO (E), A

i

PB(R), we call X

1

(A

1

)

+¯+X

n

(A

n

) a

history effect and this corresponds to the history ((X

1

,A

1

),...,(X

n

,A

n

)) of the history structure

(S (E),O (E), P). For N

i

PP (R), the observable on E with value space (R

n

,A(N

1

,...,N

n

))

induced by X

1

(A

1

)

+¯+X

n

(A

n

), A

i

PN

i

, i

51,...,n, is called a history observable and is denoted

by X

s

5X

1

+¯+X

n

. This observable corresponds to the history set s

5((X

1

,N

1

),...,(X

n

,N

n

))

on (S (E),O (E), P). If

a

PS (E), then

a

(X

1

(A

1

)

+¯+X

n

(A

n

)) is the probability of the corre-

sponding history in the state

a

and the distribution of X

s

gives the joint sequential distribution for

the history set s in the state

a

. It is also clear that an observable automorphism

c

Paut

O

(E)

induces an automorphism on the history structure (S (E),O (E), P). Moreover, a dynamical group
on E induces a dynamical group on (S (E),O (E), P).

It is of interest to note that a fixed history set s corresponds to a classical logic in the sense that

the range R(X

s

)

#E of X

s

is a Boolean algebra.

7

In this Boolean algebra, the operations

∨

and

∧

are computed within R(X

s

) and not relative to the total effect algebra E.

~In general E is not even

a lattice.

! In fact, if X

s

(A),X

s

(B)

PR(X

s

), then X

s

(A)

8

5X

s

(A

8

) and

X

s

~A!

∨

X

s

~B!5X

s

~AøB!,

X

s

~A!

∧

X

s

~B!5X

s

~AùB!.

The ‘‘local’’ Boolean algebra R(X

s

) is contextual in the sense that it depends on s. If t is another

history set then there is no relationship between R(X

s

) and R(X

t

) in general even when s and t

both involve the same observables.

V. HILBERT SPACE HISTORIES

Let H be a complex Hilbert space. A linear operator T on H that satisfies O

<T<I is called a

Hilbert space effect

17,18

and the set of Hilbert space effects on H is denoted E(H). For S,T

PE(H) we write S'T if S1TPE(H) and in this case we define S

%

T

5S1T.

Lemma 5.1: The system (E(H),O,I,

%

) is a

s

-effect algebra and every state

a

on E(H) has

the form

a

(T)

5tr(WT) where W is a unique density operator on H.

5784 J. Math. Phys., Vol. 39, No. 11, November 1998 Stanley Gudder

Copyright ©2001. All Rights Reserved.

Proof: It is easy to show that (E(H),O,I,

%

) is an effect algebra. To show that E(H) is a

s

-effect algebra, suppose that T

i

PE(H) with T

1

<T

2

<¯ . A standard result

19

shows that there

exists a T

PE(H) such that

^

Tx,x

&

5lim

^

T

i

x,x

&

for every x

PH. That is lim T

i

5T in the weak operator topology. Since

^

T

i

x,x

&

<

^

Tx,x

&

for

every x

PH, we conclude that T

i

<T, i51,2,... . Moreover, if SPE(H) with T

i

<S, i51,2,...,

then

^

T

i

x,x

&

<

^

Sx,x

&

for every x

PH, iPN. Hence,

^

Tx,x

&

<

^

Sx,x

&

for every x

PH so T<S.

Thus, S

5

∨

T

i

so E(H) is a

s

-effect algebra. It is well known that every state on E(H) has the

given form.

17,18

h

Because of Lemma 5.1, we can identify the set S (H) of density operators on H with the set

of states on E(H) and it is clear that S (H) is order determining. Moreover, the observables on
E (H) are given by positive operator valued

~POV! measures

17,18

on H and we denote the set of

observables by O (H). For S,T

PE(H) define S+T5S

1/2

TS

1/2

. Since

^

S

1/2

TS

1/2

x,x

&

5

^

TS

1/2

x,S

1/2

x

&

<

^

S

1/2

x,S

1/2

x

&

5

^

Sx,x

&

<

^

x,x

&

for every x

PH, we conclude that S+TPE(H).

Theorem 5.2: The system (E(H),O,I,

%

,

+) is a temporal effect algebra.

Proof: We only need to show that

+ is a temporal product. It is clear that O+T5O and T+I

5T for all TPE(H). If T

1

,T

2

PE(H) with T

1

'T

2

, then

S

+~T

1

%

T

2

!5S+~T

1

1T

2

!5S

1/2

~T

1

1T

2

!S

1/2

5S

1/2

T

1

S

1/2

%

S

1/2

T

2

S

1/2

5S+T

1

%

S

+T

2

.

Now suppose that T

i

PE(H) with T

1

<T

2

<¯ . As in the proof of Lemma 5.1, lim T

i

5

∨

T

i

in

the weak operator topology. Moreover, since S

+T

1

<S+T

2

<¯ ,lim S+T

i

5

∨

(S

+T

i

) in the weak

operator topology. Hence, for every x

PH, we have

lim

^

S

+T

i

x,x

&

5lim

^

T

i

S

1/2

x,S

1/2

x

&

5

^

~

∨

T

i

!S

1/2

x,S

1/2

x

&

5

^

S

+

∨

T

i

x,x

&

.

Thus,

S

+

∨

T

i

5lim S+T

i

5

∨

~S+T

i

!

so T

°S+T is countably additive. h

Since E(H) is a temporal effect algebra, we can form the corresponding history structure

(S (H),O (H), P) as in Sec. IV. For X

i

PO (H), A

i

PB(R), i51,...,n, we have the history effect

h

5X

1

~A

1

!+¯+X

n

~A

n

!5X

1

~A

1

!

1/2

¯X

n

21

~A

n

21

!

1/2

X

n

~A

n

!X

n

21

~A

n

21

!

1/2

¯X

1

~A

1

!

1/2

and for a state W

PS (H) the probability of h becomes

P

W

~h!5tr~WX

1

~A

1

!+¯+X

n

~A

n

!!5tr~X

n

~A

n

!

1/2

¯X

1

~A

1

!

1/2

WX

1

~A

1

!

1/2

¯X

n

~A

n

!

1/2

!.

These expressions are given in Ref. 7. For

s

5~~X

1

,N

1

!,...,~X

n

,N

n

!!PH

ˆ

we have the history observable X

s

5X

1

+¯+X

n

with value space (

R

n

,A

s

) and for W

PS (H) the

joint sequential distribution of X

1

,...,X

n

with partitions N

1

,...,N

n

is the probability measure on

(

R

n

,A

s

) given by P

W,s

(X

s

(

•))5tr(WX

s

(

•)). If S,TPE(H) and WPS (H) with tr(WS)Þ0,

then the conditional probability of T given S is

P

W

~TuS!5

tr

~WS+T!

tr

~WS!

5

tr

~S

1/2

WS

1/2

T

!

tr

~WS!

.

Hence, the conditional state W

uS

PS (H) is given by

5785

J. Math. Phys., Vol. 39, No. 11, November 1998 Stanley Gudder

Copyright ©2001. All Rights Reserved.

W

uS

5

S

1/2

WS

1/2

tr

~WS!

.

We now illustrate some manifestations of quantum interference discussed in Sec. II. Let s

5((X

1

,N ),(X

2

,N ))

PH

ˆ where N 5

$

A

1

,A

2

%

, A

2

5A

1

8

. The possible alternatives in this his-

tory set are the four history effects

X

s

~A

i

3A

j

!5X

1

~A

i

!+X

2

~A

j

! i, j51,2.

In general

X

1

+X

2

~R3A

1

!5X

s

~R3A

1

!5X

s

~A

1

3A

1

øA

2

3A

1

!5X

s

~A

1

3A

1

!1X

s

~A

2

3A

1

!

5X

1

~A

1

!+X

2

~A

1

!1X

1

~A

2

!+X

2

~A

1

!

5X

1

~A

1

!

1/2

X

2

~A

1

!X

1

~A

1

!

1/2

1X

1

~A

2

!

1/2

X

2

~A

1

!X

1

~A

2

!

1/2

ÞX

2

~A

1

!.

Thus, the marginal distribution need not coincide with the one-variable distribution. Of course,
this is just an illustration of the nonadditivity in the first argument (a

%

b)

+cÞa+c

%

b

+c.

As another example, let s

5((X

1

,N

1

),(X

2

,N

2

)), t

5((X

1

,M

1

),(X

2

,M

2

)) and suppose

that A,B

PN

1

, A

øBPM

1

, C

PN

2

ùM

2

. Then

X

s

~AøB3C!5X

s

~A3CøB3C!5X

s

~A3C!

%

X

s

~B3C!5X

1

~A!+X

2

~C!

%

X

1

~B!+X

2

~C!

X

t

~AøB3C!5X

1

+X

2

~AøB3C!5X

1

~AøB!+X

2

~C!5@X

1

~A!

%

X

1

~B!#+X

2

~C!.

These two expressions need not coincide. Hence, if W

PS (H) in general we have

P

W,s

~AøB3C!ÞP

W,t

~AøB3C!.

We thus see that refining a partition can change the probability of measurement results. Finally,
the fact that Bayes’ formula

P

W

~TuS!P

W

~S!5P

W

~SuT!P

W

~T!

does not hold in general follows from the fact that S

+T5T+S need not hold. That is, in general

S

1/2

TS

1/2

ÞT

1/2

ST

1/2

.

This framework encompasses the formalism of quantum evolution. Let U: H

→H be a unitary

operator and define

f

: E(H)

→E(H) by

f

(T)

5U

*

TU. Then

f

is an automorphism on E(H).

Indeed, it is straightforward to show that

f

is a

s

-isomorphism and to show that

f

preserves the

temporal product we have

f

~S+T!5U

*

S

1/2

TS

1/2

U

5U

*

S

1/2

UU

*

TUU

*

S

1/2

U

5~U

*

SU

!

1/2

U

*

TU

~U

*

SU

!

1/2

5

f

~S!+

f

~T!.

Let U(

t

)

5e

2i

tK/h

be a unitary evolution on H where K is the Hamiltonian of the system. It is

clear that U(

t

) is a dynamical group on E(H). If X

PO (H) is an observable at time zero, then the

corresponding observable at time

t

is

X

t

~•!5

f

t

~X~•!!5U~

t

!

*

X

~•!U~

t

!.

Thus, X

t

,

t

PR, is a quantum stochastic process. For times

t

1

,

t

2

,•••,

t

n

and A

1

,...,A

n

PB(R) a temporal history effect is given by h5X

t

1

(A

1

)

+¯+X

t

n

(A

n

). To simplify the notation,

let U

i

5U(

t

i

) and U

i, j

5U(

t

j

2

t

i

) for i

, j. We then have

h

5U

1

*

X

~A

1

!

1/2

U

1

¯U

n

21

*

X

~A

n

21

!

1/2

U

n

21

U

n

*

X

~A

n

!U

n

U

n

21

*

X

~A

n

21

!

1/2

U

n

21

¯U

1

*

X

~A

1

!

1/2

U

1

5U

1

*

X

~A

1

!

1/2

U

2,1

*

¯U

n

21,n22

*

X

~A

n

21

!

1/2

U

n,n

21

*

X

~A

n

!U

n,n

21

X

~A

n

21

!

1/2

3U

n

21,n22

¯U

2,1

X

~A

1

!

1/2

U

1

.

5786 J. Math. Phys., Vol. 39, No. 11, November 1998 Stanley Gudder

Copyright ©2001. All Rights Reserved.

If W

PS (H) is the state at time zero and

t

0

,

t

1

is the initial time under consideration, then the

corresponding state at time

t

0

is W

t

0

5U(

t

0

)

*

WU(

t

0

). The probability of h for the state W

becomes

P

W

t

0

~h!5tr~U~

t

0

!

*

WU

~

t

0

!h!

5tr~WU

1,0

*

X

~A

1

!

1/2

U

2,1

*

¯U

n

21,n22

*

X

~A

n

21

!

1/2

¯U

2,1

X

~A

1

!

1/2

U

1,0

!

and this is the form given previously in Ref. 7.

In order to compare our formalism to those given previously, we can reformulate it in the

following way. For

s

5~~X

1

,N

1

!,...,~X

n

,N

n

!!PH

ˆ

we use the notation N

5(N

1

,...,N

n

) and for A

i

PN

i

, i

51,...,n, we write A5(A

1

,...,A

n

)

PN . For APN we define the class operator

C

s

~A!5X

n

~A

n

!

1/2

X

n

21

~A

n

21

!

1/2

¯X

1

~A

1

!

1/2

.

Notice that

C

s

~A!

*

C

s

~A!5X

1

~A

1

!+¯+X

n

~A

n

!,

which is a history effect. For W

PS (H), we define the decoherence function D

s

W

: N

3N →C by

D

s

W

~A,B!5tr~WC

s

~A!

*

C

s

~B!!.

Then the following properties are easy to check.

~D1! D

s

W

(A,A)

5P

W

(X

1

(A

1

)

+¯+X

n

(A

n

))

>0;

~D2! D

s

W

(A,B)

5D

s

W

(B,A)

*

;

~D3! (

A

PN

D

s

W

(A,A)

51.

We say that s is a consistent history set relative to W if Re D

s

W

(A,B)

50 for AÞB.

We shall not pursue consistent history sets here because they have been thoroughly studied in

the literature. Moreover, the decoherence function D

s

W

is a specific Hilbert space construct and its

physical significance is not completely clear. For this reason, it is not a natural concept to intro-
duce into our axiomatic framework for a history structure or a temporal effect algebra. Neverthe-
less, we shall make some observations about D

s

W

that do not seem to have been previously

considered.

For simplicity, assume that N

5(N

1

,...,N

n

) consists of finite partitions N

i

, i

51,...,n

~which is frequently done in this formalism! and form the Hilbert space

l

2

(N )

5

$c

:N

→C

%

with inner product

^

c

,

f

&

5

(

A

PN

c

~A!

f

~A!

*

.

Now D

s

W

induces a linear operator

~which we also denote by D

s

W

! on

l

2

(N ) given by

~D

s

W

c

!~A!5

(

B

PN

D

s

W

~A,B!

c

~B!.

Applying

~D2! we have

5787

J. Math. Phys., Vol. 39, No. 11, November 1998 Stanley Gudder

Copyright ©2001. All Rights Reserved.

^

D

s

W

c

,

c

&

5

(

A,B

PN

D

s

W

~A,B!

c

~B!

c

~A!

*

5

(

A,B

PN

tr

~WC

s

*

~A!C

s

~B!!

c

~B!

c

~A!

*

5tr

F

W

(

A

PN

~

c

~A!C

s

~A!!

*

(

B

PN

c

~B!C

s

~B!

G

>0.

Hence, D

s

W

is a positive operator on

l

2

(N ). It follows from

~D3! that the trace of D

s

W

is 1.

Hence, D

s

W

is a density matrix so D

s

W

PS (

l

2

(N )). It is easy to check that W

°D

s

W

is a con-

tinuous affine map from S (H) to S (

l

2

(N )) so it has a unique continuous linear extension L

from the trace class operators on H to the trace class operators on

l

2

(N ). The adjoint transfor-

mation L

*

is a continuous linear transformation from O (

l

2

(N )) into O (H). With a little care,

these ideas can be generalized to the case in which N is not finite

~N is always countable!. We

conclude that each history set s possesses its own Hilbert space

l

2

(N ) and observables on

l

2

(N ) can be transferred in a natural way to observables on H. In a dual manner, states on H can

be transferred to states on

l

2

(N ) by employing the decoherence function. This may be a useful

physical interpretation for the decoherence function.

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5788 J. Math. Phys., Vol. 39, No. 11, November 1998 Stanley Gudder

Copyright ©2001. All Rights Reserved.