There has been a
particularly clear treatment of time travel in the context of quantum
mechanics by David Deutsch (see Deutsch 1991, and Deutsch and Lockwood
1994) in which it is claimed that quantum mechanical considerations show
that time travel never imposes any constraints on the pre-time travel
state of systems. The essence of this account is as follows.
A
quantum system starts in state S1, interacts with its older self, after
the interaction is in state S2, time travels while developing into state
S3, then interacts with its younger self, and ends in state S4 (see
figure 10).
Figure 10
Deutsch assumes that the set of possible
states of this system are the mixed states, i.e. are represented by the
density matrices over the Hilbert space of that system. Deutsch then
shows that for any initial state S1, any unitary interaction between the
older and younger self, and any unitary development during time travel,
there is a consistent solution, i.e. there is at least one pair of
states S2 and S3 such that when S1 interacts with S3 it will change to
state S2 and S2 will then develop into S3. The states S2, S3 and S4 will
typically be not be pure states, i.e. will be non-trivial mixed states,
even if S1 is pure. In order to understand how this leads to
interpretational problems let us give an example. Consider a system that
has a two dimensional Hilbert space with as a basis the states and .
Let us suppose that when state of the young system encounters state of
the older system, they interact and the young system develops into state
and the old system remains in state . In obvious notation:
1 3 develops into 2 4.
Similarly, suppose that:
1 3 develops into 2 4,
1 3 develops into 2 4, and
1 3 develops into 2 4.
Let
us furthermore assume that there is no development of the state of the
system during time travel, i.e. that 2 develops into 3, and that 2
develops into 3.
Now, if the only possible states of the system were
and (i.e. if there were no superpositions or mixtures of these states),
then there is a constraint on initial states: initial state 1 is
impossible. For if 1 interacts with 3 then it will develop into 2,
which, during time travel, will develop into 3, which inconsistent with
the assumed state 3. Similarly if 1 interacts with 3 it will develop
into 2, which will then develop into 3 which is also inconsistent. Thus
the system can not start in state 1.
But, says Deutsch, in quantum
mechanics such a system can also be in any mixture of the states and .
Suppose that the older system, prior to the interaction, is in a state
S3 which is an equal mixture of 50% 3 and 50% 3. Then the younger system
during the interaction will develop into a mixture of 50% 2 and 50% 2,
which will then develop into a mixture of 50% 3 and 50% 3, which is
consistent! More generally Deutsch uses a fixed point theorem to show
that no matter what the unitary development during interaction is, and
no matter what the unitary development during time travel is, for any
state S1 there is always a state S3 (which typically is not a pure
state) which causes S1 to develop into a state S2 which develops into
that state S3. Thus quantum mechanics comes to the rescue: it shows in
all generality that no constraints on initial states are needed!
One
might wonder why Deutsch appeals to mixed states: will superpositions of
states and not suffice? Unfortunately such an idea does not work.
Suppose again that the initial state is 1. One might suggest that that
if state S3 is 1/ 3 + 1/ 3 one will obtain a consistent development. For
one might think that when initial state 1 encounters the superposition
1/ 3 + 1/ 3, it will develop into superposition 1/ 2 + 1/ 2, and that
this in turn will develop into 1/ 3 + 1/ 3, as desired. However this is
not correct. For initial state 1 when it encounters 1/ 3 + 1/ 3, will
develop into the entangled state 1/ 2 4 + 1/ 2 4. In so far as one can
speak of the state of the young system after this interaction, it is in
the mixture of 50% 2 and 50% 2, not in the superposition 1/ 2 + 1/ 2. So
Deutsch does need his recourse to mixed states.
This clarification
of why Deutsch needs his mixtures does however indicate a serious worry
about the simplifications that are part of Deutsch's account. After the
interaction the old and young system will (typically) be in an entangled
state. Although for purposes of a measurement on one of the two systems
one can say that this system is in a mixed state, one can not represent
the full state of the two systems by specifying the mixed state of each
separate part, as there are correlations between observables of the two
systems that are not represented by these two mixed states, but are
represented in the joint entangled state. But if there really is an
entangled state of the old and young systems directly after the
interaction, how is one to represent the subsequent development of this
entangled state? Will the state of the younger system remain entangled
with the state of the older system as the younger system time travels
and the older system moves on into the future?
On what space-like
surfaces are we to imagine this total entangled state to be? At this
point it becomes clear that there is no obvious and simple way to extend
elementary non-relativistic quantum mechanics to space-times with
closed time-like curves. There have been more sophisticated approaches
than Deutsch's to time travel, using technical machinery from quantum
field theory and differentiable manifolds (see e.g. Friedman et al 1991,
Earman and Smeenk 1999, and references therein). But out of such
approaches no results anywhere near as clear and interesting as
Deutsch's have been forthcoming.
How does Deutsch avoid these
complications? Deutsch assumes a mixed state S3 of the older system
prior to the interaction with the younger system. He lets it interact
with an arbitrary pure state S1 younger system. After this interaction
there is an entangled state S of the two systems. Deutsch computes the
mixed state S2 of the younger system which is implied by this entangled
state S . His demand for consistency then is just that this mixed state
S2 develops into the mixed state S3. Now it is not at all clear that
this is a legitimate way to simplify the problem of time travel in
quantum mechanics. But even if we grant him this simplification there is
a problem: how are we to understand these mixtures?
If we take an
ignorance interpretation of mixtures we run into trouble. For suppose
that we assume that in each individual case each older system is either
in state 3 or in state 3 prior to the interaction. Then we regain our
paradox. Deutsch instead recommends the following, many worlds, picture
of mixtures. Suppose we start with state 1 in all worlds. In some of the
many worlds the older system will be in the 3 state, let us call them
A-worlds, and in some worlds, B-worlds, it will be in the 3 state. Thus
in A-worlds after interaction we will have state 2 , and in B-worlds we
will have state 2. During time travel the 2 state will remain the same,
i.e turn into state 3, but the systems in question will travel from
A-worlds to B-worlds. Similarly the 2 states will travel from the
B-worlds to the A-worlds, thus preserving consistency.
Now whatever
one thinks of the merits of many worlds interpretations, and of this
understanding of it applied to mixtures, in the end one does not obtain
genuine time travel in Deutsch's account. The systems in question travel
from one time in one world to another time in another world, but no
system travels to an earlier time in the same world. (This is so at
least in the normal sense of the word ‘world’, the sense that one means
when, for instance, one says "there was, and will be, only one Elvis
Presley in this world".) Thus, even if it were a reasonable view, it is
not quite as interesting as it may have initially seemed.
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