The Foundations of Quantum Mechanics and The Limitations of Human Being

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1 The Foundations of Quantum Mechanics and The Limitations of Human Beings Department of Mathematics 21 February 2011 Supported by NSF

2 Quantum mechanics is very successful in computing quantities than can be compared to experimental data: spectral lines, energies of chemical bonds, interference patterns,... What we know: We have rules for predicting the outcome of any experiment the quantum formalism. What is controversial: What actually happens inside atoms, or in quantum experiments.

The quantum formalism, part 1 (rules for predicting the outcome of any experiment) A system of N electrons/protons/neutrons has a wave function ψ t : R 3N C ψ t (q) = ψ t (q 1,..., q N ) color = phase.

3 The quantum formalism, part 1 (rules for predicting the outcome of any experiment) A system of N electrons/protons/neutrons has a wave function ψ t : R 3N C ψ t (q) = ψ t (q 1,..., q N ) color = phase. Picture: B. Thaller. As long as the system is isolated, the wave function changes with time according to the Schrödinger equation i ψ t t = N k=1 2 2m k 2 ψ t + V ψ t, which determines ψ t at any time t if the initial wave function ψ t0 any other time t 0 is given. at

4 The quantum formalism, part 2 Measurement postulate : Suppose we let the system interact with a macroscopic apparatus at time t. Formula for the probability that the apparatus obtains the outcome z: Prob(z) = Pz ψ t (q) 2 dq. P z are appropriate projection operators associated with this type of apparatus. A = z zp z is a self-adjoint operator, called the observable. Collapse of the wave function : After the interaction, ψ must be replaced by P z ψ. Example before: after: system of N = 1 particle apparatus = detector outcome = position of the particle Prob(q) = ψ t (q) 2 Collapse:

of waves), the wave passing through will show alternating fringes of brightness and darkness an interference pattern (consequence

5 An example: the double-slit experiment, part 1 Brightness = ψ 2, color = phase. Pictures: B. Thaller. Let a wave function of wave length λ hit a screen with two narrow slits (width λ). Due to interference (i.e., partial cancellation of waves), the wave passing through will show alternating fringes of brightness and darkness an interference pattern (consequence of the Schrödinger equation). Behind the double-slit, let the wave function hit a detector screen: we obtain a random spot with probability distribution ψ(q) 2.

6 An example: the double-slit experiment, part 2 Repeat the experiment many times to obtain many spots on the detection screen with distribution ψ 2.

7 The traditional orthodox Copenhagen view: One should not ask the question What happens?. Werner Heisenberg in 1958: The idea of an objective real world whose smallest parts exist objectively in the same sense as stones or trees exist, independently of whether or not we observe them [...], is impossible. We can no longer speak of the behavior of the particle independently of the process of observation. Well, the impossibility claim is actually not true. W. Heisenberg ( )

8 Three quantum theories without observers Bohmian mechanics Many-worlds (with adequate ontology) GRW theory of spontaneous collapse Each of these theories can be understood as clearly as classical mechanics, and each of them makes predictions (identical or extremely close) to those of the quantum formalism. There may be theoretical or philosophical reasons for preferring one to the others.

9 John S. Bell: In 1952 I saw the impossible done. It was in papers by David Bohm. Bohm showed explicitly how parameters could indeed be introduced, into non-relativistic wave mechanics, with the help of which the indeterministic description could be transformed into a deterministic one. More importantly, in my opinion, the subjectivity of the orthodox version, the necessary reference to the observer, could be eliminated. Bohmian mechanics: takes the word particle literally: The k-th particle has position Q k (t) R 3 at time t. The complete description of a system is (Q 1,..., Q N, ψ). The equation of motion is the simplest possible: dq k dt = probability current probability density with current = ( /m k ) Im ψ k ψ and density = ψ 2. David Bohm ( )

10 Bohmian mechanics takes wave particle dualism literally: there is a wave, and there is a particle. The path of the particle depends on the wave. Shown: A double-slit and 80 possible paths of Bohm s particle. The wave passes through both slits, the particle through only one. So in Bohmian mechanics, the particle is not in two places at the same time.

11 If the configuration Q = (Q 1,..., Q N ) has probability distribution ψ 0 2 initially (as we assume from now on) then it has distribution ψ t 2 at any time t. The quantum formalism is a theorem in Bohmian mechanics: It exactly summarizes what the inhabitants of a Bohmian universe will see.

12 Most paths arrive where ψ 2 is large that s how the interference pattern arises.

13 Current research on Bohmian mechanics Since Bohm adds an equation (and not just philosophy), one can do mathematical research on Bohmian mechanics. Some sources of open problems are: extend Bohmian mechanics to relativistic quantum mechanics, to quantum field theory, and to quantum gravity Work by Bell and by Tumulka et al. about a variant of Bohmian mechanics that allows for particle creation and annihilation. Configuration space (e.g.) N=0 R3N, actual configuration Q(t) follows a stochastic (Markov) process. open: how to define Bohm-like trajectories for photons open: quantum gravity mathematical analysis of the defining equations existence and uniqueness of trajectories Q(t) for all t and almost all Q(0) [Goldstein et al., Tumulka et al.] on a curved space-time manifold with singularities (as in the Schwarzschild space-time) with particle creation/annihilation establish the classical limit of Bohmian mechanics explain the boson fermion symmetrization postulate

Theory 2: many worlds Schrödinger proposed this theory in 1925: Matter is distributed continuously in space with density m(q, t) = N k=1 R 3N δ 3 (q q k ) ψ t (q) 2 dq.

14 Theory 2: many worlds Schrödinger proposed this theory in 1925: Matter is distributed continuously in space with density m(q, t) = N k=1 R 3N δ 3 (q q k ) ψ t (q) 2 dq. Schrödinger soon abandoned this theory because he thought it made wrong predictions. But actually, it is a many-worlds theory making right predictions: it implies the quantum formalism. E. Schrödinger ( ) For Schrödinger s cat, ψ = 1 2 ψ dead ψ alive, it follows that m = 1 2 m dead m alive. There is a dead cat and a live cat, but they are like ghosts to each other (they do not notice each other), as they do not interact. So to speak, they live in parallel worlds.

Many worlds Not knowing about Schrödinger s proposal, Everett advocated a many-worlds view in 1957, but with an unclear (or inadequate) ontology: His idea was that for wave functions such as

16 Many worlds Not knowing about Schrödinger s proposal, Everett advocated a many-worlds view in 1957, but with an unclear (or inadequate) ontology: His idea was that for wave functions such as Schrödinger s cat s, both cats are in the wave function, so both cats exist. Everett contributed substantially to the analysis of probabilities in a many-world framework. Hugh Everett ( ) Today, Everett s view is perhaps the most popular view among physicists.

17 Theory 3: GRW theory of spontaneous collapse There are different versions of the Ghirardi Rimini Weber (= GRW) theory. I present the version GRWf using the flash ontology.

Another ingredient of the GRW theory Idea: The Schrödinger equation is only an approximation, valid for systems with few particles (N < 10 4 ) but not for macroscopic systems (N > 10 23 ).

86 Another ingredient of the GRW theory Idea: The Schrödinger equation is only an approximation, valid for systems with few particles (N < 10 4 ) but not for macroscopic systems (N > ). The true evolution law for the wave function is non-linear and stochastic (i.e., inherently random) and avoids superpositions (such as Schrödinger s cat) of macroscopically different contributions. Put differently, regard the collapse of ψ as a physical process governed by mathematical laws. GianCarlo Ghirardi (born 1935) Explicit equations by Ghirardi, Rimini, and Weber (1986)

87 GRW s stochastic evolution for ψ is designed for non-relativistic quantum mechanics of N particles meant to replace Schrödinger eq as a fundamental law of nature involves two new constants of nature: λ sec 1, called collapse rate per particle. σ 10 7 m, called collapse width. Def: ψ evolves as if an observer outside the universe made, at random times with rate Nλ, quantum measurements of the position observable of a randomly selected particle with inaccuracy σ. rate Nλ means that prob(an event in the next dt seconds) = Nλ dt. more explicitly: Schrödinger evolution interrupted by jumps of the form ψ T + = e (q k q) 2 4σ 2 ψ T, i.e., multiplication by a Gauss function with random label k, center q and time T. A flash occurs at (T, q) for each collapse.

88 GRW s spontaneous collapse before the spontaneous collapse : and after:

89 The predictions of the GRW theory deviate very very slightly from the quantum formalism. At present, no experimental test is possible (but may be possible by 2020). Some sources of research questions: extend to relativistic quantum mechanics, to quantum field theory, and to quantum gravity Work by Bell and by Tumulka about relativistic version for non-interacting particles open: the same for interacting particles derive and analyze predictions predict deviations from QM [Ghirardi, Pearle, Bassi] derive a GRW formalism parallel to the quantum formalism [Tumulka et al.] mathematical analysis of the defining equations limitations to knowledge (Tumulka, NSF grant # SES )

90 Limitations of human beings Limitations to knowledge that apply equally to all human beings, animals, aliens, computers, other machines, and future inventions.

91 Limitations to knowledge in Bohmian mechanics, part 1 Theorem (Goldstein et al., 1992) For the inhabitants of a hypothetical universe governed by Bohmian mechanics, it is impossible to know the position of a particle more precisely than the ϕ 2 distribution allows, where ϕ is the (conditional) wave function of the particle. ϕ(x) = ψ(x, Y ), where Q = (X, Y ) When measuring the particle s position X with inaccuracy ε, then ϕ collapses to a narrower fct of width ε.

92 Limitations to knowledge in Bohmian mechanics, part 2 Theorem in Bohmian mechanics, and a theorem in ordinary QM You cannot measure a particle s wave function: There is no experiment that could be applied to any given particle with unknown wave function ψ and would determine ψ (with any useful reliability and accuracy). For example, in H = C 2 there is a 2-parameter family of ψs (with ψ = 1 and modulo global phase), but (it can be shown) any experiment yields essentially just 1 bit of outcome. If you are given N >> 1 particles, each with wave fct ψ, then you can determine ψ to arbitrary accuracy and reliability if N is sufficiently large. If you know that a certain particle has wave fct ψ then you can prove it, in the following sense: You can specify an experiment (with observable P Cψ ) that yields outcome 1 with prob. 1 and 0 with prob. 0; if you didn t know ψ the prob. of 0 would be positive. Upshot: Nature can keep a secret. She knows what the wave function is, but doesn t allow us to measure it.

93 Limitations to knowledge in Bohmian mechanics, part 3 Theorem (Goldstein et al., 2004) In a Bohmian universe, there is no experiment that would measure the velocity of any given particle with unknown wave function. (If ψ is known, measure Q and compute Im ψ(q)/ψ(q).) Some people dislike Bohmian mechanics for that reason. (That s a poor reason, given that one can t measure ψ either.) Bell in 1987: To admit things not visible to the gross creatures that we are is, in my opinion, to show a decent humility, and not just a lamentable addiction to metaphysics. John S. Bell ( )

94 Limitations to knowledge in GRW theory, part 1 Theorem (Tumulka et al., 2007) Also in a GRWf world, you can t measure the wave function of single given system.

95 Limitations to knowledge in GRW theory, part 2 Conjecture 1 In a GRWf world you can t reliably measure, with microscopic accuracy, the space-time pattern of flashes, or even only the number C of flashes (= number of collapses) in a given system in a given time interval [t 1, t 2 ]. I.e., there is no Geiger counter for collapses. Here, we also demand that the interaction with the measuring device does not substantially alter the pattern of flashes in the system during [t 1, t 2 ]. (Otherwise, a procedure creating flashes may pass for one measuring flashes.)

96 Limitations to knowledge in GRW theory, part 3 Example 1: Suppose we know that ψ(t 1 ) = 1 2 ψ here ψ there, (1) and H = 0. Suppose that P(C = 0) = 1 2 = P(C = 1). If C = 1 then 1 ψ(t 2 ) = ψ here or ψ(t 2 ) = ψ there, each with prob. 2. Experiment: To get information about C, we wait until t 2, then do a quantum measurement of P Cψ(t1). If outcome Z = 0 then C = 1; if Z = 1 then P(C = 0 Z = 1) = 2 3. Conjecture 2 Even when the initial wave function is known, it is impossible to determine with 100% reliability whether a collapse has occurred or not. Indeed, no other experiment can retrodict C for the initial wave function (1) with greater reliability than the experiment above.

97 Limitations to knowledge in GRW theory, part 4 Conjecture 3 If no information about the system s initial wave function is given, then no experiment can reveal any information at all about the value of C (= number of flashes during [t 1, t 2 ]). That is: Without any experiment, we know that C has Poisson distribution with expectation Nλ(t 2 t 1 ). Given the outcome Z of any experiment, the conditional distribution P(C Z) will not be narrower. In Example 1, the experiment did reveal useful information, but the initial ψ was known.

98 Thank you for your attention

On the Present Status of Quantum Mechanics

On the Present Status of Quantum Mechanics Department of Mathematics UNAM, Mexico City, 24 September 2015 What does the cat example mean? It s often called a paradox, but that is too weak that sounds like get used to it. Basically, it s an argument: