| Essays |  |
1. Definition of terms and endeavor
1.1 Unpredictability versus Indeterminacy
Consider a pendulum and assume we describe its state s(t) at time t solely by the position of its
bob . For simplicity suppose also that time is quantized in multiples of the basic time unit 1.
If kept unperturbed, the pendulum’s motion is periodic ; thus any state s(t) will be be achieved
infinitely many times. Assume for instance that s(t1)=s(t2). Does this imply s(t1+1)=s(t2+1) ?
Clearly, there can be no such implication, since we do not know the velocity of the bob at the
two times in question. Therefore, the system as described here is indeterministic. We might,
however, imagine a description of the state in terms of more parameters ; with respect to this
more extensive characterization the system might turn out to be deterministic. In that
case, if we were given the parameters q1,q2, q3,...qn-1 qn,we would be able to predict
the state s(q1,q2, q3,...qn-1 qn, t+1) at time t+1 from our knowledge of the state
s(q1,q2, q3,...qn-1 qn,t) at time t.
1.2 The difficulty of establishing non-determinism
The example illustrates the difference between unpredictability and non-determinism.
If insufficiently characterized, the pendulum’s motion is unpredictable. However,
this does not preclude the existence of a complete description s(q1,q2, q3,...qn-1 qn,t)
of the pendulum’s states that will make the system deterministic. Unfortunately this means
that it is very difficult to prove that a system is truly non-deterministic. Proving that
a process is deterministic requires the more straightforward (though possibly very difficult)
task of specifying the complete set of descriptive parameters for a state together with the
rules by which we can obtain one state from the preceding ; proving that a process is
non-deterministic, however, requires us to prove that no such complete description and
set of rules can be found, whether in practice or in theory.
Due to this difficulty, our philosophy of physics professor would be unlikely to ask
his students to write an essay discussing whether or not the world behaves deterministically,
or, whether or not an accurate deterministic description of the world could be found. What we
can discuss, is whether or not, given some theory X, the acceptance of X as a correct
description of the world entails that the world described by this theory X behaves
deterministically.
1.3 Assumptions made
All discussion in this essay is based on the assumption that we take Quantum Mechanics
to be correct ( an assumption that seems reasonable, as QM is right now the physical
theory that is most accurately verified by experiment ) ; we then explore if this assumption
forces us to conclude that the world is non-deterministic. Note that calling Quantum Mechanics «
correct » is not equivalent with calling it « complete » . If I say that we take
Quantum Mechanics to be correct, it is in the weakest sense possible, namely in the
sense of « not false ». In this weakened sense, the assumption seems tenable,
as none of the interpretations and standpoints to be discussed in this paper
contradict the validity of Quantum Mechanics. If Bohm proposes that the
probabilistic nature of Quantum theory can be complemented by additional
parameters and equations so as to yield a deterministic theory, then he is
not showing that the theory of Quantum Mechanics was wrong, but merely that it
was not the most complete theory possible, since we had probabilities in places
where according to Bohm certainty can be achieved. Considering that Schroedinger
dynamics as we know it constitutes an essential part of Bohm’s theory and that the
wave equation evolves in time just as in standard QM, makes clear that Bohm is
very far from refuting the correctness of QM. Similarly, if the EPR paradox seems
to show that Quantum Theory is an incomplete description of the world, this does
not mean any of the equations in our theory are false. It would imply only that
the description of the world by quantum mechanics is insufficient, i.e that
the states and evolution of states in the world are not exhaustively
characterized by the models of our theory.
2. If we accept Quantum Mechanics, must we conclude the world is non-deterministic ?
2.1 Interpreting QM, rethinking the world
Suppose we consider QM to be a correct physical theory about our world. Then in order to discuss
whether or not this implies that the world is deterministic, requires us to examine two main
questions, the first of which seems –elusively- trivial . The questions are
1. Is Quantum Mechanics deterministic ?
2. Does Quantum Mechanics furnish a complete description of the world ?
The first question could seem self-evident if considered quickly. Without deeper reflection,
most people would probably find it « beyond doubt » that Quantum Mechanics is not deterministic.
How could a theory that at best gives us a choice or a spread of different outcomes and their
probabilities ever be called deterministic ? However, whether Quantum Mechanics is considered
to be deterministic, depends sensitively on our notion of what there is to be determined, i.e.
what we take to be the elements of the physical world that we endeavor to describe by our
physical theory. If we take the definite outcomes and measurements of experiment to be the
constituents of reality, then clearly Quantum Mechanics as it stands right now is not
deterministic. This is because even though the states in the unmeasured system follow the
deterministic evolution of Schroedinger dynamics, Quantum Mechanics offers no account as
to what determines the actual outcome upon measurement, i.e. what determines the one
eigenstate in the quantum mechanical superposition to which the system collapses.
Taking the world to be a system very close to our notions of experiment and measurable
definiteness, however, is not mandatory. We could equally well defend a picture in which
the real world is not taken to correspond to our notion of definite values of observable
quantities, but instead is taken as a world in which only quantum mechanical wave functions
and other mathematical constructs are elements of reality (of course, such an interpretation
is faced with the task to at least try to explain the reason for our « definite » perception,
i.e. our notion that things have definite values, and our inability to ever measure something
not corresponding to an eigenstate of the observable in question.) In conjunction with such
a « wave function world » , Quantum Mechanics is perfectly deterministic because the
Schroedinger equation gives us all that we need in order to predict with certainty the
state s( t+∆t) at some time t+∆t from our knowledge of the state s(t) at
time t (note that we are by no means predetermining definite values of observables,
as the very notion of « the value of an observable » is a non-sensical term in this
particular world view).
The above distinctions lead me to assign the different interpretatons and ideas to be
discussed in the next section into two categories : a) Those ideas and interpretations
that rethink the theory of quantum mechanics, b) those interpretations of Quantum Mechanics
that really are reinterpretations of the world.
2.2 Different Interpretations
a) Interpretations/ critiques that rethink the theory :
I assume familiarity with the Standard, or Orthodox, Interpretation of Quantum Mechanics
(the most famous proponent of which was Niels Bohr). This interpretation, which can be found
in any Physics textbook and which is the one most commonly entertained by Physicists, is both
simple and messy at once. It is elegant in the sense that it fits all the « data » without
overloading itself with too many additional structures, such as multiple worlds (as in MWI) ,
additional formulae (as in Bohmian Mechanics) or new randomly governed processes (as in GRW).
On the other hand it leaves unsolved the puzzle of measurement (When does a measurement take
place ? What is a measurement ?) and the dilemma of the two dynamics :
a « non-measurement » Schroedinger dynamics, which is continuous and deterministic on
the one hand, and a measurement dynamics characterized by the non-deterministic collapse
of the wave function into one of the system’s eigenstates on the other hand.
The Standard Interpretation was economic and straightforward enough as to satisfy most working
physicists. It was also messy and disturbing enough to start a chain reaction of ever new
interpretations of quantum mechanics during the following decades that tried to fix the
measurement problem, i.e. 1. the problem of how to determine whether or not a measurement
is taking place and 2. the problem associated with the randomness of the collapse.
GRW, a theory put forth by Ghirardi, Riminini, and Weber, represents an attempt to solve
the first problem, that is the problem of deciding which of the two dynamics
( the Schroedinger or the Measurement dynamics) should be used when. GRW solves this
problem by turning both these dynamics into one. Schroedinger dynamics is still taken
to govern the evolution of our quantum states as represented by the wave functions, at
least most of the time. The remaining time, i.e. some short instants separated by long
time intervals, is governed randomly. The leisurely deterministic Schroedinger wave
evolution is thus interrupted by random and sudden localizations of the wave functions.
The definite results of a measurement are then explained by means of these random
localizations and the fact that macroscopic measurement devices consist of a multitude
of different, entangled quantum systems that collapse/ become localized as soon as
one of them does. Although GRW solves the problem of the two dynamics, it clearly
does not alleviate the uneasiness of having a random element in our quantum theory,
but in fact brings in additional randomly governed processes (i.e. the random
localizations). In this sense GRW is clearly not deterministic. Thus, if we were
to assume GRW is correct and in that it is a complete description of the world,
and if we furthermore assumed that definite measurement outcomes are in fact
true elements of physical reality, we would have to conclude that Quantum
Mechanics (as seem from the GRW perspective) forces us give up on the
determinism of the world.
Another interpretation, one that tries to fix both the problem of randomness and the problem
of two dynamics, was advanced by Bohm in 1952. In what came to be known as « Bohmian Mechanics »
, Bohm describes how the specification of the positions of a system, along with an additional
formula, the so-called Guiding Equation, could complement the traditional formalism of
Quantum Mechanics so as to turn it into a deterministic theory of the world. Problems
with this theory are that it is non-local and also that it ascribes a superior position
to the observable « position », which in this view is taken to be physically « more real »
than other observables, such as momentum or spin. This introduces numerous formal
difficulties in dealing with the theory. Opponents of this theory have also voiced
concern because this theory breaks the symmetry between momentum and position that
we cherished in our original quantum mechanical theory. It is important to note,
however, that the latter, though not irrelevant, does not qualify as rigorous
scientific critique, as (unfortunately) nowhere is written a law stating that
nature shall always be symmetric (or as symmetric as possible). Nevertheless,
it represents a valid proposal that has not proven to be incorrect and that
boldly illustrates the possibility of serious proposals for deterministic
quantum theories, a fact that should have pleased Einstein, who insisted that
« God does not play dice. » In fact, if Bohm’s theory could be verified,
it would once and for all settle the question of determinism and lead us
to conclude that the world is deterministic. In that lucky case we would
not even have to give up our intuition of what constitutes the true elements
of reality and would not have to accept the reality of the wave function
over the reality of definite measurement results, a rethinking of the
world that is described further in part b).
b) Interpretations that rethink the world :
In my opinion a salient characteristic of the interpretations of QM that rethink the world is that
they all seem isomorphic to each other. I will describe the Many Worlds Interpretation, the
Many Minds Interpretation, and lastly, Mermin’s Ithaca Interpretation of Quantum Mechanics ,
which I find to be the most beautiful formulation among these. The Many Worlds Interpretation
is a non-collapse theory. For simplicity, we illustrate the theory using the case of
observables that admit discrete eigenvalues. Consider a quantum mechanical state of
superposition s = ∑civi, (with sigma summing from 1 to m) where the vα are
eigenstates of the system, the ci complex scalars. MWI abandons the collapse postulate
by asserting that the unmeasured superposition state s does not collapse into one of
the eigenstates vi. Rather, MWI claims that if we conduct an experiment with m different
possible outcomes vi , then each outcome occurs with certainty, but in its own world ;
thus, if i ranges from 1 to m, the world will split into m different worlds, each
corresponding to a unique outcome vi and a measurable value λi that is the
eigenvalue corresponding to vi. The greatest defect of this theory it at once
its greatest advantage. This defect lies in the impossibility of ever testing
the theory experimentally, a fact that is nicely illustrated in the entry
« The Many-World Interpretation of Quantum Mechanics » ( I assume their
use of the word collapse to stand for collapse in the MWI sense, i.e.
the splitting into different worlds upon performing a quantum experiment)
of the Stanford Encyclopedia of Philosophy:
« To observe the collapse we would need a super technology, which allows ‘undoing’ a quantum
experiment, including a reversal of the detection process by macroscopic devices. [...] These
proposals are all for gedanken experiments that cannot be performed with current or any foreseen
future technology. Indeed, in the experiments an interference of different worlds has to be
observed. »
Thus the theory of MWI cannot -- at least not in any experimental, most likely not even in
any theoretical way—be proven or disproven, a feature that can be held both for and against it.
Whether or not this theory is deterministic is a tricky question. Randomness is eliminated
in the sense that, given a set of possibilities, whether or not one outcome is going to occur
over another is not governed randomly. All possible outcomes occur with certainty, i.e. with
probability 1. Outcomes that were not quantum mechanically possible outcomes for the experiment
at hand have probability 0. Seeing that everything gets assigned P=1 or P=0, should we
conclude that we are witnessing a deterministic theory in action ? On the one hand, we
seem to be certain about outcomes that will happen. But in a way we are « too certain »,
so much, in fact (excuse the sarcasm), that if we add the probabilities of the different
outcomes we obtain an overall probability m >1 ! I claim, therefore, that within the
context of MWI discussions about probability, determinism or randomness are void of
meaning. Of course this also entails that it is a vacuously true statement to say that
we cannot prove the theory to be non-deterministic.
I take the Many Minds Interpretation (MMI) to be mostly just a reformulation of MWI that
is isomorphic to the former. Instead of talking about the splitting of worlds, each accomodating
a specific experimental outcome, MMI claims the mind of the experimenter to undergo a
superposition that in a way « parallels » the superposition state of the unmeasured quantum
system. Each summand in our mind state’s linear combination corresponds to an outcome of the
experiment, each component of our mind believing to have obtained a definite measurement
result.
The mathematically prettiest interpretation, in my opinion, is the Ithaca Interpretation of
Quantum Mechanics by N. David Mermin, which is very well explained in his recent paper « What
is Quantum Mechanics Trying to Tell Us ? « (1998). While our notion of definite perception
and measurement finds analogs within the theories of MWI and MMI, Mermin’s theory liberates
itself completely from the need for such structures. In MWI our « definite outcomes
» corresponded to the definite outcomes in each of the many worlds that are created by
the splitting ; in MMI the definite outcomes corresponded to summands contained in our
mind’s linear combination. IIQM simply negates the physical significance, even existence
of these outcomes. Mermin summarizes his own theory as follows: « Correlations have
physical reality ; that which they correlate does not [...] And that’s all there is
to it. The rest is commentary. »
I believe that this interpretation of the world can be considered deterministic, because if
probabilistic correlations are physically real, while the correlated observables are not, then
the probabilistic nature of QM need not trouble us. Mermin explains why the probabilistic nature
of QM need not worry us anymore :
.
« If correlations constitute the full content of physical reality, then the fundamental role
probability plays in quantum mechanics has nothing to do with ignorance. The correlata – those
properties we would be ignorant of—have no physical reality. There is nothing for us to be
ignorant of. »
2. 3 More than one way to save determinism
The above considerations show that the acceptance of quantum mechanics does not in itself
necessitate the acceptance of a non-deterministic world.
As mentioned in part 1, establishing the indeterminacy of the world would require us to prove
the non-existence of a correct, complete theory (that could be similar to quantum mechanics,
but also completely different) which determines the evolution of systems in our world ; using
the more formal definition of determinim by van Fraassen , we would have to prove that there
exists no « dynamic group « of evolution operators Uα satisfying Ubv(t)=v(t+b) and
UbUc = Ub+c for any state v(t) and any positive real numbers b and c. I believe such a
non-existence proof to be impossible both in practice and in principle. Even if we were
able to show that Quantum Mechanics could not be extended (or slightly altered) in order
to become a deterministic theory, this would not rule out that another (possibly very different) complete deterministic theory might exist.
On the other hand, several of the above-mentioned interpretations would, if accepted, entail
the determinism of the physical world.
Bohm’s theory, for instance, if taken to be correct, would imply that the world as we perceive
and measure it is indeed a deterministic system. If, given the initial position and state of
a particle, we are nevertheless unable to fully predict subsequent states, the reason for
this is the fact that by measuring an observable, we inevitably interfere with the system and perturb the quantity to be measured (this is some sort of « hands-on » uncertainty relation, to be distinguished from Heisenberg’s uncertainty principle, which independently of measurement follows from the mathematical structure of the theory and the requirement that only eigenvalues can be measured, which means that only commuting and hence mutually diagonizable observables can be simultaneously measured with complete accuracy). The underlying dynamics, however, is deterministic and, moreover, assigns not wavy spreads, but definite values to the positions of a particle at later times.
But even if we find ourselves profoundly suspicious of Bohm’s theory-- a theory that after all
seems to mix classical, definite « positions » with quantum mechanical waves in one big
soup-- we can still « save« the determinism of the world, if we wish. Namely, we can
redraw our image of what we believe to be the elements of the real physical world around us.
If we conclude, as in the Many Minds Interpretation or the Bare Theory, that the actual
constituents of physical reality are wave functions, while our classical everyday
perception is merely an illusion , then quantum mechanics provides a complete,
deterministic description of this world, and thus the world, entirely
characterized
by QM, is deterministic. It will evolve continuously and leisurely, obeying
at all times the Schroedinger equation HΨ=EΨ.
Why then have I heard several times in the past few weeks the surprised comment
(by people not taking Phi 327, of course) « Why, of course Quantum Mechanics forces
us to give up determinism ! ». I believe that this (mis)conception follows the following
chain of reasoning :
A : Quantum Mechanics is correct AND QM provides a complete description of the world.
B : The real world and the elements of reality correspond to the definite outcomes we
perceive and measure in physical experiment (that is our world consists of things such
as positions, momenta, etc, not of wave functions and superpositions of dead and alive cats.)
Then, A and B together imply that the world as we empirically identify it is indeterministic.
The dicussion in this paper showed that both assumptions A and B are not evident and can be put
to debate. Quantum Mechanics, even if considered to be correct, might be incomplete. B need
not be an adequate definiton of « the world » as the world can be defined in other ways than
the world corresponding to our common perception of definite measurements and perception .
Thus I am not saying the above reasoning is necessarily wrong. In its raw form, however,
the argumentation is not stringent because it rests on assumptions that lie at the heart of
the debate in the philosophy of Quantum Mechanics.
3.The Lost, the Rescued, and what happened with free will
3.1 Trading intuition/perception for mathematical beauty ?
As indicated amply in this paper, if asked whether or not the acceptance of QM forces up
to give up determinism, I would respond with a definite « No ». This, surprisingly, did
not require that I adopt one particular interpretation out of the myriad of options available.
For the purpose of building my argument it was enough to show that interpretations consistent
with a deterministic world exist and are tenable. If asked, however – and this does not touch
the central argument or conclusion of this essay- to name my favorite picture, I would opt
for one of the « bare » theories, such as MMI of IIQM. These theories, although they reshape
our entire picture of reality and thus might seem far from intuitive, appeal to me because
of their mathematical beauty. Such a world would retain the symmetries described by van
Fraassen and be rigorously describable by the purest of linear algebra and its more general
mathematical siblings. It would furthermore discard aspects of the other interpretations
that I personally find puzzling and not on a deep level comprehensible, such as the
element of randomness involved in GRW’s spontaneous localization or the random collapse
of the wave function into one of its eigenstates as pictured in the Copenhagen
interpretation (If one « small » thing is random, why isn’t everything ? Granted
that, in GRW, Schroedinger dynamics serves as a kind of « deterministic glue
» that is active during the time intervals lying between the random localizations,
wouldn’t we have to know also how states get entangled and disentangled in order
to establish that the system won’t « go crazy » ? For if entanglement was also
random, then it seems that the indeterminacy in GRW would not just be some
sort of « well-behaved » randomness, but in fact a randomness that would
blow up so strongly as to render the system acausal) . Thus I hold not
that this theory is the most sensible, but that it is the prettiest.
3.2 Happy Physicists ?
By chosing a bare theory, we seem to have sacrificed our intuition regarding the consistence
of the physical world, as we probably all have trouble envisioning not a world of objects having
positions and momenta, but instead a world of wave functions, a large algebraic spread of
mathematical constructs that do not find an equivalent in the world as we perceive it, and,
even if we forced the comparison far, would at most lead to absurd images of cats that are
simultaneously dead and alive, and of students that are in a superposition of attending their
philosophy of physics lecture and taking a vacation in Europe. Apart from mathematical beauty,
then, does this viewpoint gain us anything ? « Determinism ! » you might excitedly yell, but
after a while you would realize that our joy of this is not self-evident. One might say
that this interpretaiton would allow physicists to find what they were after, a clue
indicating that indeed « God does not play dice » (as Einstein’s famous quote goes) and
that the endeavor of finding laws of nature is not a futile one, as indeed nature is
deterministic and thus its behavior, at least in theory, expressable by laws. However,
this joy might be spoiled by the realization that we have to give up the perceived and
measured quantities of our everyday experience as elements of the real world, which
might deeply trouble one or the other experimental physicist. Sloppily said,
physicists found determinism (and thus hope), but lost the world (as we know it).
3.3 Free Will : Not a claim, just a disclaimer.
As I have already trespassed the rules of objective reasoning by having opted for the bare
interpretaiton of quantum mechanics based primarily on matters of mathematical taste, I
would like to adress one more conviction that I have found to influence people’s standpoint
on whether or not the world is deterministic. A friend told me the other day that he personally
would like determinism to be true because he is fond of free will. Instead of discussing this
problem at length I would only like to ask a question which is raised by Jean Bricmont in his
paper « Determinism, Chaos and Quantum Mechanics ». My question to the adherents of free
will is : Is (pre-)determined will worse than random will ?
4. Why all this ?
I believe that the discussion of this paper and in the philosophy of quantum mechanics in general
illustrates the importance of interpretations of quantum mechanics both for the revision and
development of new physical theories about the world, as well as for our comprehension of
what the world is, i.e. what kind system our theories are trying to describe in the
first place. Many physicists, however, do not seem to take this endeavor seriously and it
is sad that the worst account of interpretive problems in quantum mechanics that I have
read in the last few weeks is written in the appendix of my Physics department Quantum
Mechanics course. Physicists like Henry Margenau, former professor at Yale, who in his book
« The Nature of Physical Reality » stresses the importance of philosophical considerations
as integral part of the physical discipline, still appear to be a minority. And
frequently we hear comments, as the following one
(quoted in the article « Event-Enhanced Quantum Theory and Piecewise Deterministic
Dynamics » by Ph. Blanchard and A. Jadczyk ) :
« We can be sitting there and discussing [...] philosophical implications and
the deep questions
of quantum physics while the computer is cranking out numbers which we need for practical
purposes and which we could never obtain in any other way. What more can we ask for ? »
A lot more, I believe, and I hope more physicists will.
Bibliography :
Albert, David Z. : Bohm’s Alternative to Quantum Mechanics.
Scientific American May 1994
Blanchard and Jadzyk : Event-Enhanced Quantum Theory and Piecewise Deterministic
Dynamics. arXiv :hep-th/9409189 v1 29 Sep 1994
Bricmont, Jean : Determinism, Chaos and Quantum Mechanics.
http://www.fyma.ucl.ac.be/files/Turin.pdf
Van Fraassen, Bas C.: Laws and Symmetry.
Clarendon Press. Oxford. 1989
Griffiths, David J. : Introduction to Quantum Mechanics.
Prentice Hall. 1995.
Margenau, Henry : The Nature of Physical Reality.
McGraw-Hill Book Company. 1950.
Mermin, N. David : What is Quantum Mechanics Trying to Tell Us ?
arXiv : quant-ph/9801057 v Sep 1998
Smart, J.J.C : Between science and philosophy.
Random House. New York. 1968.
Stanford Encyclopedia of Philosophy :
Bohmian Mechanics.
http://plato.stanford.edu/entries/qm-bohm/
Many-Worlds Interpretation of Quantum Mechanics
http://plato.stanford.edu/entries/qm-manyworlds/
Earman, John : A Primer On Determinism. D. Reidel Publishing Company. 1986.
1 The example is taken from Bas C. van Fraassen : Laws and Symmetry
2 described in Mermin, David N. : What is Quantum Mechanics Trying to Tell Us ?
3 p. 255 van Fraassen : Laws and Symmetry