Introducing the divinity of the Universe
to pave the way for scientifically credible theology
Chapter 12: Is Hilbert space independent of Minkowski space?
Synopsis
In previous chapters I have proposed a theological picture, identifying the initial singularity with the omnipotent Christian God of pure actuality developed by Aquinas from the work of Aristotle. We imagine the first structure to emerge in this singularity to be the Hilbert space of quantum mechanics which provides the variation necessary for evolution. As in biological evolution, only those entities able to reproduce themselves are selected for survival. In this chapter I present evidence that quantum mechanics is independent of spacetime and creates the formal elements of all the visible features of the Universe, including ourselves, which it contains.
Contents
12.1 Do Lorentz transformations affect Hilbert space?
12.2 Entanglement: infinite velocity or no space-time?
12.3 The Lagrangian view of quantum mechanics
12.4: Quantum non-locality
12.1: Do Lorentz transformations affect Hilbert space?
12.2 Entanglement: infinite velocity or no space-time?
12.3 The Lagrangian view of quantum mechanics
12.4: Quantum non-locality
12.1: Do Lorentz transformations affect Hilbert space?
Since the advent of special relativity, classical physical theories are which involve velocities comparable to the speed of light are usually described in flat Minkowski space-time. The special principle of relativity holds that every observer sees the same laws of physics, including the same speed of light, in their own rest frame. This defines the Lorentz transformation which is expressed succinctly by the 1, 1, 1, -1 metric of Minkowski space.
If we set the speed of light c to 1, all observers see an invariant interval ds2 = dx2 + dy2 + dz2 - dt2. It seems to be generally accepted in quantum field theory that the Lorentz transformation applies equally to states in Hilbert space and to particles in Minkowski space. This implies that Minkowski space is the domain of Hilbert space.Since in quantum mechanics observables are represented by hermitian operators which act on the Hilbert space of state vectors one expects the analogue in relativistic quantum mechanics of a classical observable field to be a set of hermitian operators defined at each point in spacetime and having a well defined transformation law under the appropriate group. Streater & Wightman (2000): PCT, Spin, Statistics and All That
What is the reason for this? Perhaps it is just familiarity. Physics has always been a matter of space and time. Why would it ever be different? There may also be Newton's theological feeling that space and time were part of God's creation. In the General Scholium he writes:
And from his true dominion it follows that the true God is a Living, Intelligent, and Powerful Being; . . . He endures forever, and is every where present; and, by existing always and every where, he constitutes Duration and Space. Since every particle of Space is always, and every indivisible moment of Duration is every where, certainly the Maker and Lord of all things cannot be never and no where.
12.2: Entanglement: infinite velocity or no space-time?
If the quantum world constitutes a layer of the Universe built on the initial singularity before the emergence of observable energy, time, space and momentum, Hilbert space would become the domain of Minkowski space. The phenomenon of entanglement suggests that the Hilbert quantum world exists prior to and independent of the Minkowski classical world. It seems more reasonable to attribute the apparently instantaneous long distance propagation of correlations associated with entanglement (spooky action at a distance) to the absence of space rather than to the unrelativistic propagation of these correlations at infinite velocity.
If this is the case, there opens up a new degree of freedom in the relationship between quantum and classical dynamics which may make it possible to remove some of the confusion noted by Kuhlmann in his critique of quantum field theory:
In conclusion one has to recall that one reason why the ontological interpretation of QFT is so difficult is the fact that it is exceptionally unclear which parts of the formalism should be taken to represent anything physical in the first place. And it looks as if that problem will persist for quite some time. Meinard Kuhlmann: Quantum Field Theory
Einstein appears never to have been truly happy with quantum mechanics, and often sought to demonstrate its weaknesses. In the course of a paper intended to show that quantum mechanics is incomplete, the authors (E, P & R) identified spooky action at a distance which is a now seen as a consequence of entanglement. Einstein, Podolsky and Rosen: Can the Quantum Mechanical Description of Physical Reality be Considered Complete?
EPR argue from the quantum formalism that a measurement on one part of an entangled system enables us to predict with certainty the outcome of a measurement on the other system even though they are spacelike separated. They concluded that 'no reasonable definition of reality could be expected to permit this.' Careful experiments have, however, shown that entanglement is real.
Here we may exemplify it and its consequences using a simple two state system. Electrons have two spin states, often called up and down. In the singlet state, one electron has spin up, the other down, so that the total spin of the singlet is zero.
Entanglement establishes that when these two electrons spatially separated, they behave, when observed, as though they are still in contact. If one electron is observed to be spin up, the other will be observed to be spin down no matter how far apart they are. This is 'spooky action at a distance', but the fact that this correlation appears to be instantaneous suggests that although the electrons are distant in Minkowski space, they are still in contact in Hilbert space. If this is the case, it is a major break from conventional wisdom and opens the way for an interesting new approach to theology, spirituality and quantum field theory.
Although the biggest shock that came with quantum mechanics is the inability to predict the precise timing of events, entanglement gives us a very definite method predict both the nature and timing of entangled events, since the observation of one part of an entangled system appears to have an immediate consequence for the other part.
Experimental tests of entanglement have gradually improved. It has been shown that this correlation at a distance operates at many times the velocity of light. Salart, Baas, Branciard, Gisin & Zbinden (2008): Testing the speed of 'spooky action at a distance
We might therefore think of the Universe at the Hilbert space level as being a work in progress. The uncertainty of quantum behaviour helps to make the evolution of the Universe possible by providing the variation necessary for creative evolution (see Chapter 13: The emergence of quantum mechanics).
12.3: The Lagrangian view of quantum mechanics
I am trying to preserve as much as possible of the old theology of Aristotle and Aquinas and of modern physics as I follow my dream of showing that the Universe fulfils all the traditional roles attributed to the God. Aquinas defined God as pure action, a term he inherited from Aristotle. A modern definition of action was developed by Joseph-Louis Lagrange. Lagrange's new formulation of Newtonian mechanics make it easier to deal with many body problems like the whole solar system. Lagrangian mechanics - Wikipedia
In 1933, Paul Dirac felt that it would be advantageous if Hamilton's principle could also be applied to quantum mechanics. As Dirac notes, one of the features of the Lagrangian method is that it allows one to collect together all the equations of motion and express then as the stationary property of a certain action function. A principal role of quantum mechanics is to find stationary points known as eigenvalues in the motion of quantum systems. The second desirable feature is that the action function is a relativistic invariant. This is consistent with the idea that action exists prior to the emergence of spacetime, so that relativistic transformations do not affect it. At first sight it did not seem clear to Dirac how the Lagrangian could be introduced to quantum mechanics. The natural home of classical relativistic Lagrangian mechanics is Minkowski space: The equations involve partial derivatives of the spacetime coordinates and velocities, and no meaning can be given to such derivatives in quantum mechanics. So he thought that he needed to take over the ideas of the classical Lagrangian theory, not the equations of the classical Lagrangian theory. What he discovered is that from the Lagrangian point of view quantum mechanics deals only with the kinematic time and phase element of events in spacetime. Hamilton's principle is equivalent to the quantum mechanical search for stationary phases which are integral multiples of the quantum of action, the quantum of angular momentum. P. A. M. Dirac (1933): The Lagrangian in Quantum Mechanics He found that the Lagrangian could be applied to quantum theory if he were is restrict the treatment of classical action in 4-dimensional spacetime to the single timelike dimension which correlates with the evolution of phase in quantum mechanics. The Lagrangian then appears as the exponent in a complex representation of the evolution of the phase of a quantum system. Stationary action in the classical world appears as fixed phase in the quantum regime. This might slsdo accout for the relativistic invariance of the Langrangian. This seems to connect with the role of Hermitian operators in quantum theory which select stationary eigenvectors out of the ceaseless motion of quantum systems described by the Schrödinger equation. One of the important symmetry principles in physics is known as local symmetry. All the modern field theories for the fundamental interactions use this symmetry. Local symmetry is now called gauge invariance, but as Chen Ning Yang points out, if we were to rename it today we would call it phase invariance and gauge fields would be called phase fields. Feynman used this idea to create his path integral representation of quantum mechanics. He was able to show this to be equivalent to the Schrödinger and Heisenberg representations of the theory. The path integral has since become the representation of choice in the introduction of quantum mechanics into quantum field theoretical and attempts to produce a quantum theory of gravitation. Feynman's representation also supports the idea, proposed here, that Hilbert space is prior to and independent of Minkowski space. The path integral approach assumes that a particle moving from a to b follows every possible path between these two points. Its quantum phase is integrated along every one of these paths and the sum of these integrals computed to produce an amplitude for the transition from a to b. The absolute square of this amplitude yields the probability of the transition. Given the size of the space-time universe and the relativistic restriction on the speed of particle motion, this computation can only make sense in a Hilbert space where spacetime is absent. We can imagine that the path integral approach selects paths which, like a Bohr orbit in an atom, involve a stationary integral number of quanta of action. The overall program on this site is that we should replace continuous arithmetic (which leads to infinities and other problems) with logic and interpret the quantum of a action as a logical operator. Such operators are discrete rather than continuous, which is consistent with the quantum nature of the world. Finally, the independence of Hilbert and Minkowski space is supported by the radical differences in their metrics, space-time distance and the speed of light in Minkowski space, phase or rate of change of phase in Hilbert space.12.4: Quantum non-locality
The experiments referred to above have demonstrated that entangled particles can act upon one another at a distance even though their separation is spacelike. This is called quantum non-locality.
Classical physics is founded on a belief in local realism which has three features:
1. regularities in observed phenomena point to the existence of physical reality independent of human observers;
2. consistent sets of observations underlie 'inductive inference', the notion that we can use them to devise models of what is going on behind the scenes; and
3. causal influences cannot travel faster than the velocity of light.
Long experience and detailed argument has shown that quantum mechanics is not a local realistic theory. The EPR argument was perhaps the first hint that local realism is false. John Bell studied EPR and formulated a first version of Bell's theorem which would show that quantum mechanics was not a local realistic theory. The phenomena and theory both appear to point to the fact that the quantum Hilbert world is prior to and independent of the relativistic Minkowski world. Myrvold, Genovese & Shimony (Stanford Encyclopedia of Philosophy): Bell's Theorem
There is further discussion relationship between Hilbert and Minkowski space in Chapter 20: Measurement: the interface between Hilbert and Minkowski spaces.
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Notes and references
Further reading
Books
Streater (2000), Raymond F, and Arthur S Wightman, PCT, Spin, Statistics and All That, Princeton University Press 2000 Amazon product description: 'PCT, Spin and Statistics, and All That is the classic summary of and introduction to the achievements of Axiomatic Quantum Field Theory. This theory gives precise mathematical responses to questions like: What is a quantized field? What are the physically indispensable attributes of a quantized field? Furthermore, Axiomatic Field Theory shows that a number of physically important predictions of quantum field theory are mathematical consequences of the axioms. Here Raymond Streater and Arthur Wightman treat only results that can be rigorously proved, and these are presented in an elegant style that makes them available to a broad range of physics and theoretical mathematics.'
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Links
Einstein, Podolsky and Rosen, Can the Quantum Mechanical Description of Physical Reality be Considered Complete?, A PDF of the classic paper. 'In a complete theory there is an element corresponding to each element of reality. A sufficient condition for the reality of a physical quantity is the possibility of predicting it with certainty, without disturbing the system. In quantum mechanics in the case of two physical quantities described by non-commuting operators, the knowledge of one precludes the knowledge of the other. Then either (1) the description of reality given by the wave function in quantum mechanics is not complete or (2) these two quantities cannot have simultaneous reality. Consideration of the problem of making predictions concerning a system on the basis of measurements made on another system that had previously interacted with it leads to the result that if (1) is false then (2) is also false, One is thus led to conclude that the description of reality given by the wave function is not complete.' back |
Lagrangian mechanics - Wikipedia, Lagrangian mechanics - Wikipedia, the free encyclopedia, ' Introduced by the Italian-French mathematician and astronomer Joseph-Louis Lagrange in 1788, Lagrangian mechanics is a formulation of classical mechanics and is founded on the stationary action principle. Given a system of point masses and a pair, t1 and t2 Lagrangian mechanics postulates that the system's trajectory (describing evolution of the system over time) . . . must be a stationary point of the action functional S = ∫ L dt. By convention, L = T − V, where T and V are the kinetic and potential energy of the system, respectively.' back |
Meinard Kuhlmann (Stanford Encyclopedia of Philosophy), Quantum Field Theory, ' Quantum Field Theory (QFT) is the mathematical and conceptual framework for contemporary elementary particle physics. In a rather informal sense QFT is the extension of quantum mechanics (QM), dealing with particles, over to fields, i.e. systems with an infinite number of degrees of freedom. (See the entry on quantum mechanics.) In the last few years QFT has become a more widely discussed topic in philosophy of science, with questions ranging from methodology and semantics to ontology. QFT taken seriously in its metaphysical implications seems to give a picture of the world which is at variance with central classical conceptions of particles and fields, and even with some features of QM.' back |
Myrvold, Genovese & Shimony (Stanford Encyclopedia of Philosophy), Bell's Theorem, ' Beginning in the 1970s, there has been a series of experiments of increasing sophistication to test whether the Bell inequalities are satisfied. With few exceptions, the results of these experiments have confirmed the quantum mechanical predictions, violating the relevant Bell Inequalities. Until recently, however, each of these experiments was vulnerable to at least one of two loopholes, referred to as the communication, or locality loophole, and the detection loophole (see section 5). Finally, in 2015, experiments were performed that demonstrated violation of Bell inequalities with these loopholes blocked. This has consequences for our physical worldview; the conditions that entail Bell inequalities are, arguably, an integral part of the physical worldview that was accepted prior to the advent of quantum mechanics. If one accepts the lessons of the experimental results, then some one or other of these conditions must be rejected.' back |
P. A. M. Dirac (1933), The Lagrangian in Quantum Mechanics, ' . . . there is an alternative formulation [to the Hamiltonian] in classical dynamics, provided by the Lagrangian. This requires one to work in terms of coordinates and velocities instead of coordinates and momenta. The two formulation are closely related but there are reasons for believing that the Lagrangian one is more fundamental. . . . Secondly the lagrangian method can easily be expressed relativistically, on account of the action function being a relativistic invariant; . . .. ' [This article was first published in Physikalische Zeitschrift der Sowjetunion, Band 3, Heft 1 (1933), pp. 64–72.] back |
Salart, Baas, Branciard, Gisin & Zbinden (2008), Testing the speed of 'spooky action at a distance, ' Correlations are generally described by one of two mechanisms: either a first event influences a second one by sending information encoded in bosons or other physical carriers, or the correlated events have some common causes in their shared history. Quantum physics predicts an entirely different kind of cause for some correlations, named entanglement. This reveals itself in correlations that violate Bell inequalities (implying that they cannot be described by common causes) between space-like separated events (implying that they cannot be described by classical communication). Many Bell tests have been performed, and loopholes related to locality and detection have been closed in several independent experiments. It is still possible that a first event could influence a second, but the speed of this hypothetical influence (Einstein's 'spooky action at a distance') would need to be defined in some universal privileged reference frame and be greater than the speed of light. Here we put stringent experimental bounds on the speed of all such hypothetical influences. We performed a Bell test over more than 24 hours between two villages separated by 18 km and approximately east-west oriented, with the source located precisely in the middle. We continuously observed two-photon interferences well above the Bell inequality threshold. Taking advantage of the Earth's rotation, the configuration of our experiment allowed us to determine, for any hypothetically privileged frame, a lower bound for the speed of the influence. For example, if such a privileged reference frame exists and is such that the Earth's speed in this frame is less than 10(-3) times that of the speed of light, then the speed of the influence would have to exceed that of light by at least four orders of magnitude.' back |