Step by step, our ideas about the nature of our reality have moved far from the sensory constructions of space and time that define our immediate experience. And once fully outside the knowledge brought with sensation, we lose our footing. It’s difficult to manage ‘what can’t be sensed.’ But our conceptual difficulties with quantum mechanics are very reasonable if we imagine that, however abstract, the mathematics that got us there is rooted in our sensory experience--our sense of space and duration, our perception of quantity, and perhaps the cognitive mechanisms that manage these. The remarkable refinement of mathematical ideas has forced a reconsideration of what we think we see, and the conceptual possibilities that mathematics provides may indicate that we’ve enhanced our sensory apparatus in such a way that it has been made sensitive enough to ‘reach’ the edge of what is ‘sensible’ making us aware of the reality that escapes us. It is beginning to look like vast parts of our reality are not sensible.

These thoughts came to mind after reading a few discussions concerning the reconciliation of quantum mechanical strangeness. A New Scientist article addresses, specifically, questions about the meaning of space and time and their place (or lack of it) in modern physics. Space and time are constructed by the body and further explored by mathematics. But questions about whether or how they are real are very old. As the article’s author Anil Ananthaswamy says:

“…arguments about the nature of space and time swirl on. Are both basic constituents of reality, or neither – or does one perhaps emerge from the other in some way? We are yet to reach a conclusive answer, but it is becoming clear that if we wish to make further progress in physics, we must. The route to a truly powerful theory of reality passes through an intimate understanding of space and time.”

I like the phrase ‘intimate understanding.’  It suggests getting very close to their source.  The difficulty in physics, more specifically, as described in the New Scientist article, is this:

“A quantum object’s state is described by a wave function, a mathematical object living in an abstract space, known as Hilbert space, that encompasses all the possible states of the object. We can tell how the wave function evolves in time, moving from one state in its Hilbert space to another, using the Schrödinger equation. In this picture, time is itself not part of the Hilbert space where everything else physical sits, but somehow lives outside it…As for space, its status depends on what you are measuring. The wave function of an electron orbiting the atomic nucleus will include properties of physical space such as the electron’s distance from the nucleus. But the wave function describing the quantum spin of an isolated electron has no mention of space: according to the mathematics, the picture we often paint of an electron physically
 rotating is meaningless.”

Although a Hilbert space is a vector space whose structure is completely abstract, it rests on the meaning that it borrows from the relationships among vectors that we imagine in two and three dimensional Euclidean space. And so it is grounded in a familiar spacial idea. But mathematics has grown in such a way that the vectors of a Hilbert space can be used to represent possible states of a quantum mechanical system.

In a recent Scientific American blog, science journalist and FQXi member George Musser gave University of Maryland philosopher Ruth Kastner the opportunity to discuss her ideas about how to resolve the difficulties with what is called the transactional interpretation of quantum mechanics. She begins with a reference to physicist and writer Han Christian von Baeyer’s article in the June issue of Scientific American. She says about his article that in response to the deep questions about the meaning of quantum theory von Baeyer “discusses one proposal -- a denial that the theory describes anything objectively real …”

This is a very misleading summary of what von Baeyer discusses. I also wrote a few weeks ago about this article. It’s the *wave function* that is thought to have no objective reality from the point of view that von Baeyer discusses. This perspective (called Qbism) combines quantum theory and probability theory and sees the wave function as a powerful mathematical tool that provides the observer a way to make decisions about the surrounding quantum world. I took note, in my post, of the fact that Bayesian statistics (the ones used in Qbism) are also used by cognitive scientists to model how we build our very immediate expectations of our world from sensory data. And I quoted physicist Christopher Fuchs, a prominent proponent of Qbism, who said:

“…even if quantum theory is purely a theory for apportioning and structuring degrees of belief, the question of “Why the quantum?” is nonetheless a question of what it is about the actual, real, objective character of the world that compels us to use this framework for reasoning rather than another.””

This doesn’t sound like a denial that the theory describes anything real.

The transactional Interpretation that Kastner discusses in her blog was first proposed, she tells us,  by physicist John Cramer in the 1980s and can be traced back to physicists John Wheeler and Richard Feynman.  She says the following:

“My development of the Transactional Interpretation makes use of an important idea of Werner Heisenberg: “Atoms and the elementary particles themselves … form a world of potentialities or possibilities rather than things of the facts.” This world of potentialities is not contained within space and time; it is a higher-dimensional world whose structure is described by the mathematics of quantum theory.”

I wasn’t able to get a very clear picture of the history of the idea, nor her development of it from the blog, but the punch line is clear enough. Her transactional picture assumes that “there is more to reality than what can be contained within space-time.” It is somehow with the encounter of “potential events” (not contained within space-time) that “real energy may be conveyed within spacetime…”  and “delivered in the normal future direction.”

She goes on to say:

“The transactional picture is conceptually challenging because the underlying processes are so different from what we are used to in our classical world of experience, and we must allow for the startling idea that there is more to reality than what can be contained within spacetime.”

I had the impulse to bring these ideas together because of what they have in common, specifically, that what we see (in the most abstract sense of that word) consistently indicates how much we don’t see. And, perhaps that there is even a strain on our mathematical ways because mathematics itself may have its roots in how the body ‘perceives.’ This doesn’t mean that progress can’t be made.  Bringing back a little bit from Christopher Fuchs:

“For the Qbist, the lesson that the structure of quantum theory calls out to be interpreted in only this way is that the world is an unimaginably rich one in comparison to the reductionist dream.  It says that the world has excitement, risk, and adventure at its very core.”

Considering how our mathematics brings us closer to this core is equally exciting.

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Joselle DiNunzio Kehoe is a writer and Lecturer of Mathematics at the University of Texas at Dallas. This post is reproduced from her blog at Mathematics Rising.

this post has been edited by the author since its original submission

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"The route to a truly powerful theory of reality passes through an intimate understanding of space and time."

Science has been going in the opposite direction for a century. Once you accept that "a sentient being can jump, "within a minute" (of his experienced time) arbitrarily far in the future, say sixty million years ahead", any further discussion of space and time is meaningless and even harmful:

Thibault Damour: "The paradigm of the special relativistic upheaval of the usual concept of time is the twin paradox. Let us emphasize that this striking example of time dilation proves that time travel (towards the future) is possible. As a gedanken experiment (if we neglect practicalities such as the technology needed for reaching velocities comparable to the velocity of light, the cost of the fuel and the capacity of the traveller to sustain high accelerations), it shows that a sentient being can jump, "within a minute" (of his experienced time) arbitrarily far in the future, say sixty million years ahead, and see, and be part of, what (will) happen then on Earth. This is a clear way of realizing that the future "already exists" (as we can experience it "in a minute")."

Pentcho Valev

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Nice article.

My own essay in this year's contest speaks to the difficulties of implementing deterministic numerical results in a probabilistic Hilbert space with no time parameter.

Going a little tangential to the subject: It continues to amaze me that many critics of string theory, which is a very soundly constructed physical theory with no mystical connotations, deride the theory's lack (so far) of experimental predictions -- while wholly accepting the self-fulfilling results of quantum experiments.

Tom

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The question is posed, is time a fundamental constituent of reality or not.

This is actually a false dilemma. My essay Programming Bits and Getting Its explains why.

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My own emphasis is not so much on the presence of unseen things, as it is on how mathematics has somehow validated the conceptual difficulties we have with what physics tells us. I think it likely that mathematics is a symbolic formalization (or externalization) of biological processes involved in perception and learning. And physics is certainly about our perceived reality. So maybe we shouldn't only ask questions about the nature of that reality, but also about what mathematics may be expressing about what it means to perceive something.

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The constants c and ħ in naturalized units are just one. The speed of light just intertwines position and time. It is a conversion factor. In addition the proper time for a photon is zero. A set of null rays through a point gives the light cone, and we can rescale these rays any way we wish and that proper time is still zero. The light cone is then a projective space in Lorentz space. This freedom to rescale the speed of light to anything we want can further be seen in most equations that involve the speed of light. If we change the Planck length or the Bohr radius of the H-atom by rescaling c the size of any ruler changes as well. This completely cancels out any rescaling of the speed of light.

The Planck constant is similar. This intertwines the uncertainty in position and momentum. Further any rescaling of ħ has the same effect on various fundamental equations involving this constant.

The gravitational constant has naturalized units of area. Other gauge charges are similar, for we can express Coulomb’s law as

F = kqq’/r^2

so that q has units of length and qq’ is area. This connection to area has a bearing on the holographic principle.

We measure c and ħ with physical objects that are defined by things like charges and so forth. The number we assign to the speed of light c = 299,997km/sec is really a manifestation of these other constants. These constants are the result of renormalization group and the hierarchy problem with unification of gauge couplings. This is an active area of research.

LC

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I quote from R. Kastner

"Nature is never perfectly symmetric."

Yes, the many apparent symmetries in our theories up to retro-causality are unnatural artifacts.

With "reversibility we would have time-symmetric (isotropic) processes, which would fail to transfer energy, preclude change, and therefore render the whole notion of time meaningless."

Well this is obvious to common sense.

I merely maintain my view that the symmetry of the laws is a result of abstraction, and continuation from traces that are non-symmetric but unilaterally extended only in past time. "In future time" refers to mere potentialities, a notion used by Peirce as to characterize real numbers in contrast to what we may call the finitist Hilbert's space of states.

Eckard, 1793

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