Gennaro Auletta* Quantum Mechanics and Contingency

Introduction It is well known that one of the biggest difficulties for cosmology is to explain why the constants of nature have the values that they actually have. In other words, why the universe has the matter and large-scale structure it actually has.1 It is also well known that this problem has to do with the existence of life and of human beings (the anthropic principle).2 The concern is understandable, since physicists look for symmetries and try to explain why there are symmetry breaks. For this reason, a theory of multiverse is born, that is the idea that actually any possible combination of the constants values is instantiated in some universe. This would represent, indeed, a huge symmetry, especially taking into account the number of these universes. A similar question arose in quantum mechanics, related with the measurement problem. Here, one could not understand why a given possible value of a measured observable is realized at random. For this reason, Everett proposed in 1957 that all values are realized simultaneously (each one for a different observer).3 Later on, DeWitt fulfil this approach by speaking of different universes, and such approach is known now as Many World Interpretation (MWI).4 This approach is also sometimes connected with the cosmological proposal of the multiverse. In particular, by making use of Tegmark's terminology5, the quantum-mechanical version of the multiverse (his level III) absorbs both the problem of initial conditions (level I) and that of the values of natural constants (level II).6 In this way the problem of the initial conditions both of our universe and of any physical experience or system be radically solved. It is a sort of supersymmetry that would completely abolish the contingency of the physical conditions.

* Pontifical Gregorian University, Rome; e-mail: auletta@unigre.it. 1 For a good summary of the problem see M. Tegmark's paper, http://www.uboeschenstein.ch/boe/themes/multiversum.html (see also http://space.mit.edu/home/tegmark/multiverse.html). 2 Barrow, J. D./Tipler, F. J., The Anthropic Cosmological Principle, Oxford, Clarendon P., 1986, 1987. Demaret, J./Lambert, D., Le principe anthropique. Lhomme est-il le centre de lunivers?, Paris, Armand Colin, 1994. 3 Everett, Hugh III, ```Relative State' Formulation of Quantum Mechanics'', Review of Modern Physics 29 (1957): 454-62, rep. in DeWitt, Bryce S./Graham, Neill, (Eds.) The Many World Interpretation of Quantum Mechanics, Princeton, University Press, 1973. 4 DeWitt, Bryce S., ``The Many-Universes Interpretation of Quantum Mechanics'', in D'Espagnat, Bernard (Ed.), Foundations of Quantum Mechanics, New York, Academic Press, 1971, rep.in DeWitt, Bryce S./Graham, Neill, (Eds.) The Many World Interpretation. 5 He affirms that parallel universes form a four-level hierarchy: They might have different initial conditions (Level I); different physical constants and particles (Level II); or different physical laws (Level IV). 6 I will not discuss here the level IV of Tegmark's proposal.

Classical and Quantum Mechanics As is well known, classical mechanics seemed to provide a completely deterministic account of our universe. It is true that the initial conditions of the dynamic evolution of a given system are mostly not controllable, neither are the conditions at contour. However, classical mechanics could solve both problems. In fact, it was assumed that, even when the initial conditions are not under control, this is only due to a limitation affecting our technology or our computation power. In principle, if one could know the antecedent conditions generating both the occurring initial conditions and the conditions at contour, one would provide a complete description of the system's dynamics. It is true that this regress could go in infinitum. However, the nature of classical-mechanical laws (in particular the fact that they are reversible and founded on the continuity of most physical quantities) allows both retrodictions and predictions from the current state of the system. This means that in principle it is possible to reach a so complete knowledge of the past and future states of the entire universe, that the problem is dissolved. A full description of the current state of our universe contains in itself the whole past and the whole future (it is both a complete memory and a complete program). In quantum mechanics, things stand in a rather different way. Here, as I have said, the outcome of a measurement is in general intrinsically random. The reason is that the system can be in a superposition state, in which the outcome is therein only in a potential way. It is not a reality in the current sense of the word but eventually becomes an actual reality upon a suitable selection due to the detection device. In this way, there is no possibility to lead such an outcome to some previous conditions which would determine it. The superposition state, indeed, cannot determine this single outcome, since, at a potential level, it comprehends not only this outcome but also possible alternatives. Let us consider a very simple example. If the initial state is a (two-dimensional) superposition of the possible outcome 0 and of the possible outcome 1 , that is, ! = c

00 + c

11 , (1)

where c0 and c

1 are some coefficients such that c

0

2

+ c1

2

=1 , then the system, as far as is in the state (1), cannot be to be in the state 0 , nor in the state 1 . This is the reason why in quantum mechanics retrodictions and predictions are in general not possible: The single outcome, say 1 , does not allow to infer that it is a causal consequence of some previous state.7 The reason for this difference between classical mechanics and quantum mechanics is the following. Though in both theories the laws ruling the behaviour of the physical systems are deterministic, in classical mechanics they rule single events and properties of the system, and for this reason provide a causal framework. In fact, we cannot speak of causality where there are not both a deterministic dynamics and a rigorous principle of determination, according to which the systems are well "localized" in space and time and posses completely determined properties. In quantum mechanics, instead, a system mostly do not exhibit determined properties but only show certain probabilities to have such properties. For instance, the outcome 1 ,

7 Wheeler, John A., ``Law Without Law'', in Wheeler, J. A./Zurek, W. (Eds.), Quantum Theory and Measurement, Princeton, University Press, 1983.

which can be associated to a property (through the corresponding projector 1 1 ), has only a certain probability to be realized if the system, which is assumed to be in the state (1), is measured, a probability given by the square modulus of the amplitude c1= 1 ! .

Therefore, the problem of the initial conditions (or, equivalently, of the final outcomes) seems irreducible in quantum mechanics. Everett's Proposal As I have said, Everett and DeWitt tried to solve such a problem by pointing out that each component of the superposition (1) is realized in a different world. In this way, there is no longer a random "choice" of a certain outcome, but any possible outcome that is contained in a superposition (in our example both 0 and 1 ) is realized. In this way, the deterministic evolution of the superposition (1), together with all quantum systems of our universe, which constitute a single world-state vector, completely determine any future (and past state), provided that we enlarge our perspective in order to comprehend any universe in which a component of some superposition state is realized. However, quantum-mechanically this proposal runs into some difficulties (as shown by Zurek8), since the most fundamental problem of the quantum measurement is not that one value is instantiated but that one observable is measured. In fact, before measuring, many possible expansions of the same state are possible, that is, many possible observables are measurable. However, these observables, due to the uncertainty relations, are in general incompatible, and therefore are not jointly measurable (this is called the degeneracy problem). In other words, any measurement requires the preliminary "choice" of a given observable. For instance, let us consider the two orthogonal state vectors !+ and !" , in terms of which 0 and 1 can be written as

0 = 12

!+ " !"( ), 1 =1

2!+ + !"( ) . (2)

It is evident that 0 and 1 are both (symmetric) superpositions of !+ and !" . This means that, if 0 and 1 are possible outcomes when measuring a certain observable, say, position, !+ and !" are possible outcomes when measuring another observable that is not commutable with that one, say, momentum. In fact, given Eqs. (2), the state (1) can also be written

! = c0 + c12

"+ +c0# c

1

2

"# . (3)

Therefore, we are forced to assume that, when measuring, we must choose between either measuring position or momentum. The question now is: What are the conditions that determine this choice?

8 Zurek, Wojciech H., ``Pointer Basis of Quantum Apparatus: Into What Mixture Does the Wave Packet Collapse?'', Physical Review D24 (1981): 1516-25

Choice and Entanglement As is well known, many quantum systems of our universe are plus or minus entangled (entanglement is indeed not an absolute concept, and there are degrees of it9).There are reasons to suppose that any quantum system is plus or minus entangled with any other one in our universe, since all systems of our universe should have a common origin in the initial big bang. This means that, when one measures a quantum system in the state (1), this, say, the particle a, can be entangled with another system, say, the particle b, such that the description of both is given by !

ab= d 0

a1

b+ d ' 1

a0

b, (4)

where d 2 + d ' 2 =1 . Note that it is impossible to ascertain the existence of such an entanglement from the local perspective of a measurement of the particle a or b (but only by comparing the results obtained by two observers dealing with measurement of systems prepared in states identical to those of a and b, respectively). It is true that: if an experimenter measures (without I knowing) the particle b and finds it to be in the state 0

b, when I measure the particle a, this must be found in the state 1

a.

However, in the mean, over a big number of particles being in the same state (4), I cannot see any difference, because the other experimenter and myself would observe these two outcomes according to the probability given by the square modulus of d'. If on the other hand, the particles I measure in a succession of trials are plus or minus entangled with many others that concur each time in a random proportion to the outcome of measurement, this conclusion is reinforced. However, if we take the MWI seriously, also the act of "choice" that we perform here is subjected to quantum-mechanical laws. And equivalently for the other experimenter. In this case, for the sake of simplicity, my state would be described as a superposition of a state !

e associated with the outcome 1

a and a state !

e

associated with the outcome 0a. Similarly, the other experimenter will be in a

superposition if state !f associated with the outcome 1

b and the state !

f

associated with the outcome 0b. Then, this description could be written

!

abef= m 0

a"

e1

b#

f+m ' 1

a#

e0

b"

f, (5)

where m 2 + m ' 2 =1 . Up to here nobody would find nothing extraordinary, since, again, there is no local anomaly. Moreover, the state (5) seems also to represent a good solution to the above problem of choice, since it correctly transfers the "choice" of the measured observable to the quantum-mechanical description of the observer. However, suppose that in our universe happens a random fluctuation, say, of the system c, which was previously in the superposition

!c=1

2

0c+ 1

c( ) , (6)

9 Vedral, V./Plenio, M. B./Rippin, M. A./Knight, P. L.,``Quantifying Entanglement'', Physical Review Letters 78 (1997): 2275-79.

such that the system jumps randomly to the state 0

c. Suppose also that this system is

entangled with the previous ones, such that !

abefc= n 0

a"

e1

b#

f0

c+ n ' 1

a#

e0

b"

f1

c, (7)

where n 2 + n ' 2 =1 . Now, the situation is a little bit worse. Since this jump is associated, for instance, to the outcome 0

a (and vice versa, since entanglement is

not a causal relation but only a non-local interdependency, and the same holds for the other involved systems). This means that, given such a random fluctuation, in our universe will never occur a situation in which 1

a is the outcome. The theoretician of

the MWI will but object: No problem, there are other universes in which 1a occurs.

This is true, but there will be no universe in which both 1a and 0

c occur (or in

which both 0a and 1

c occur). In other words, we will have a universe in which the

"line " 0

a!

e1

b"

f0

c (8)

is realized, and a universe in which the "line " 1

a!

e0

b"

f1

c (9)

is realized but neither 0

a!

e1

b"

f1

c, (10)

nor 1

a!

e0

b"

f0

c. (11)

Universes and Possibilities Since we have seen that our universe can be conceived as a huge network of plus or minus entangled system, the number of events that cannot occur together must be also huge. If we consider now, more complex situations, in which many quantum systems are involved (that is, when several configurations like (8) or (9) cross), as it is the case for mesoscopic and macroscopic bodies, it is clear that there are many state of affairs that will never realized in no universe. It is true that, according to the MWI, these bodies are a form of illusion and that only a quantum level of reality exists. However, this becomes a petitio principi so long we do not find specific reasons for dismiss what our experience (and a good part of physics) suggests.10 In other words, there are reasons to assume that not all that is abstractly possible can be realized in some universe. For this reason, Tegmark's statement according to which 10 Obviously, I will not deny that the fundamental laws of our universe are quantum-mechanical, so far we know. However, decoherence show how a macroscopic world can emerge also given such a situation.

Now you are in universe A, the one in which you are reading this sentence. Now you are in universe B, the one in which you are reading this other sentence. Put differently, universe B has an observer identical to one in universe A, except with an extra instant of memories

seems to be a little bit hazardous. Our conclusion, on the contrary, suggests that it is impossible to find an universe that is exactly the same as our one but with only one change. For this reason even if there are 10500 or 1015,000 universes, they will never cover the huge sea of all abstract possibilities, that is, of all alternatives to each single event, which all together would exhaust everything that is conceivable and therefore eliminate the problem of contingency. The root of the difficulties is in how we understand quantum superposition. If we understand this as a simple collection of possibilities, the MWI is somehow suggested. It is not by chance that Tegmark quotes David Lewis ' modal realism, that is, the idea introduced in philosophy, according to which any possible alternative to our world somehow exists.11 However, quantum superpositions do not represent possibilities. They represent rather potentialities. Unfortunately, Aristotle, who has the historical merit to have introduce the concept of potentiality, somewhere confounded it with the concept of possibility.12 Le us consider a simple example. If a quantum system is trapped in a potential well, its state is represented by the superposition of the outcomes allowed by such a situation. However, it is possible that somebody turns the potential off, in which case far more outcomes would be possible. Then, possibility is not what can happen given certain conditions, but simply what is not impossible. Therefore, possibilities have no existence at all, and they rather represent the way in which we speak of what-abstractly-is not impossible. True potentialities (the possible outcomes given the potential well) represent on the contrary a far tiny subset of the set of abstract possibilities and can be therefore considered as restrictive conditions on them. For this reason, they can be considered somehow real13, since represent a necessary condition of the outcome itself (nothing from nothing). Quantum Laws In my opinion, the physicists who have tried to see in the multiverse a final solution of the contingency problem have not reflected sufficiently on the nature of quantum laws. As I have suggested, quantum laws do not govern properties of systems but only probabilities of properties, and, for this reason, they do not rule singular events but are general in nature. As Charles Peirce already understood, any lawful explanation is 11 Lewis, David, On the Plurality of Worlds, Oxford, Blackwell, 1986. 12 Auletta, Gennaro, "Quantum Information as a General Paradigm", Foundations of Physics 35 (2005): 787-815, and "The Problems of omnimoda determinatio and Chance in Quantum Mechanics", in M. Bitbol (Ed.), Constituting Objectivity: Transcendental Approaches of Modern Physics, in press. 13 Auletta, G./Tarozzi, G., "Wavelike Correlations versus Path Detection: Another Form of Complementarity", Foundations of Physics Letters 17 (2004): 889-95 and Auletta, G./Tarozzi, G., "On the Physical Reality of Quantum Waves", Foundations of Physics 34 (2004): 1675-94.

general in nature and cannot account for the fact that, for instance, a pen is here on the desk. In his words,

Why this, which is here, is such as it is; how, for instance, if it happens to be a grain of sand, it came to be so small and so hard, we can ask; we can also ask how it got carried here; but the explanation in this case merely carries us back to the fact that it was once in some other place, where similar things might naturally be expected to be. Why IT, independently of its general characters, comes to have any definite place in the world, is not a question to be asked; it is simply an ultimate fact.14

Therefore, the MWI cannot solve the problem of the chance of quantum events, there is an irreducibility of initial conditions or final events to become absorbed in a multiversal symmetry, and contingency constitutes a irreducible feature of our universe. I hope that philosophers and theologians will work in the future in order to point out its nature. This would be a service even for science. ACKNOWLEDGMENT: I desire to warmly thank Archbishop Joseph Zycinski who inspired my research on the problem of contingency and supported it with his kind friendship.

14 The Collected Papers, Vols. I-VI (eds. Charles Hartshorne/Paul Weiss), Cambridge, MA, Harvard University Press, 1931-1935; vols. VII-VIII (ed. Arthur W. Burks), Cambridge, MA, Harvard University Press, 1958: 1.399, 1.405.