The direction of time in quantum field theory by Peter Morgan

Some may consider this paper too mathematical, although I took some trouble to keep it relatively elementary. In any case, I have submitted the paper because it was the topic of the contest that caused me to write the paper, which derives from considerations about time-reversal but gives a new perspective on quantum field theory.

It's perhaps also rather against my paper that I don't say much about the Nature of Time directly, because I'm by now in the habit of not doing too much metaphysics in papers. I believe it hinders Physicists and Mathematicians accepting new work if one makes too much of it. Nonetheless, I believe that this mathematical and conceptual approach to quantum field theory may somewhat interest those who do engage in metaphysics.

Excellent take on techniques to guarantee continuous function physics (consistent with experience and experiment) using algebraic methods. I agree in principle, though I personally think that a continuous random field must be modeled as n-dimensional continuous in order to guarantee sufficient "room" to break time reversal symmetry; i.e., a dissipative model.

Thanks for a thoughtful and well constructed essay.

Tom

Thanks, Tom.

Having looked at your FQXi Essay Contest paper again, I'm curious how your comment (your 2nd sentence) could be realized in my QFT-motivated formalism? I /think/ I don't accept the premise of your comment, however, because I take dissipation to be an achievable feature of a state over a time-reversal invariant algebra. I think a state over the algebra of observables of the continuous random field as I have constructed it here *can* be dissipative, by contingently restricting to positive-frequency test functions when constructing a state by the action of creation operators on the vacuum (or at least by a contingent time-reversal asymmetry of the state).

Something I've only realized because of responding to your comment is that the vacuum is zero energy in the continuous random field formalism -- in the sense that it is time-reversal invariant -- unlike the QFT vacuum.

That said, I think I'll riff a little. The proposal here is only that we *might* require the algebra of observables of a quantum field theory to be time-reversal invariant, in line with conventional ideas of coordinate invariance. *If* we do, in a creation and annihilation operator formalism, then the vacuum state *must* have negative frequency components as well as the usual positive frequency components, and the algebra of observables is essentially classical (in a very clear but limited sense), but the algebra of positive-frequency modes of the time-reversal invariant algebra is essentially identical to conventional quantum optics. This last fact makes it /relatively/ difficult to make an empirical argument against this new formalism.

The interesting thing, to me, is that this approach to understanding QFT by comparison with classical random fields is rather different from other approaches to the comparison of quantum/classical. I take it that continuous random fields are of interest even if we can't construct models that are as empirically effective as those of the standard model of particle physics because of the light they shine on QFT.

Hello Peter,

I enjoyed your paper!

But I was puzzled by your comment,

"It's perhaps also rather against my paper that I don't say much about the Nature of Time directly, because I'm by now in the habit of not doing too much metaphysics in papers. I believe it hinders Physicists and Mathematicians accepting new work if one makes too much of it."

I'm not sure how saying something true or new about the nature of time ought hinder hysicists and Mathematicians from accepting new work, or how talking about the nature of time would necessarily be metaphysics.

Did you know that Farady, one of Einstein's heroes, barely used math? And yet Einstein had a portrait of Faraday hanging in his office.

Lee Smolin writes in The Trouble With Physics: "Niels Bohr was an even more extreme case. Mara Beller, a historian who has ... out that there was not a single calculation in his research notebooks, " --

http://books.google.com/books?id=z5rxrnlcp3sC&pg=PA309&lpg=PA309&dq=niels+bohr's+notebooks+math&source=web&ots=SQ7ITDa2Ge&sig=XUGVG6YBP5Fk4UoccE_oh32qo40&hl=en&sa=X&oi=book_result&resnum=4&ct=result#PPA309,M1

http://www.amazon.com/gp/reader/061891868X/ref=sib_dp_srch_pop?v=search-inside&keywords=mara#

Now Bohr and Farady accomplished quite a lot, and never backed away from using words to describe physical concepts reflecting a *physical* reality.

Contrast this to the last thirty years of indecipherable maths, which say *nothing* about physics. The motivation for this physicless, wordless physics puzzles the will, until one realizes that as long as one never says anything, one can be "not even wrong" forever, guaranteeing an infinite amount of funding and a secret pass into the upper echelons of the dominant antitheory regimes--the only games in town.

But after thirty years of frozen time and frozen physics, perhaps it is time to move beyond the widely-held belief and attitude that it is somehow "uncool" to cowboy up and talk about physical reality like a man, and that we ought stop short of saying anything definitive about physical reality, but just present fancy maths instead.

Perhaps it is time to cowboy up and ride into town and liberate us all from the frozen block universe and frozen time with a simple physical postulate, expressed definitively in words, and its accompanying simple mathematical equation, which celebrates a hitherto unsung universal invariant that provides a common *physical* model for entropy, time and all its arrows and assymetries in all realms, quantum entanglement and nonlocality, and all of relativity, as well as other physical phenomena: The fourth dimension is expanding relative to the three spatial dimensions, or dx4/dt=ic:

http://fqxi.org/community/forum/topic/238

Time as an Emergent Phenomenon: Traveling Back to the Heroic Age of Physics by Elliot McGucken

Einstein stated, "Mathematics are well and good but nature keeps dragging us around by the nose. "

And so it is that we ought let Nature and physical reality lead, as Einstein, Bohr, and Farady all did, and find the math that describes it, as opposed to starting with math and then seeking out the money that funds it.

Einstein also had a sign in his Princeton office, "Everything that can be counted does not necessarily count; everything that counts cannot necessarily be counted."

A mathematician who is not also a poet will never be a complete mathematician. - Karl Weierstrass

Yes, when it comes to time, perhaps the poets from thousands of years ago can tell us more about it than the physicists who are trying to get us to forget time, space, and physics, while replacing words with math, and concrete thought with airy nothingness. But whatever time is, time is short--far too short for mere mathematical games.

Age steals away all things, even the mind.

Virgil

All our sweetest hours fly fastest.

Virgil

All things deteriorate in time.

Virgil

Better times perhaps await us who are now wretched.

Virgil

But meanwhile time flies; it flies never to be regained.

Virgil

Best,

Dr. E (The Real McCoy)

Thanks for taking a look at my paper, Peter, though I didn't mean to be self-promoting. My comment--and it was a comment, not a criticism--was motivated by the thought that Poincare recurrence in such a closed system might obviate randomization of your continuous field, and therefore present the same (mathematical) problem of time reversal symmetry as with the classical field. I agree that QFT needs the kind of mathematical approach you bring to the party, though. (An algebraic basis for the continuous field was a major part of Einstein's quest.) We differ mainly in choice of space. Obviously, I am married to supersymmetry.

Dr. E, I have to take Peter's side in the question of physics vs metaphysics. Using your own example, though Faraday was one of the greatest experimentalists ever, can one claim to truly understand his results without knowing Maxwell's equations? I don't think Einstein would disagree; special and general relativity are mathematically complete.

Tom

DrE, I rather enjoyed most of your quotes, but I didn't say that I don't do metaphysics, nor that metaphysics are not important, ... I think I said that /this/ paper is metaphysics-lite. When /I/ go heavy on the metaphysics, /I/ don't get published, but of course there are some who do metaphysics well enough to be published --- and in ways that I find interesting.

For what it's worth, however, I'm /trying/ to do Physics in a way that resists the post-positivist critique of Positivism --- which I take to be substantial, albeit, lamentably, mostly nonconstructive --- by thinking about models in Physics and "bridge principles" from those models to the world in a way that I take to be somewhat associated with Lakatos's ideas on research programs. Alternatively, I would also want to say that models /mediate/ between ourselves and the world, which switches the emphasis in what I take to be a useful but not essential way (for this, see "Models as Mediators", an edited volume by Morgan(not me)&Morrison). Taking such views seriously, and trying to be honest to whatever, vaguely, they are, makes writing for publication in Physics journals a delicate matter.

Tom, I guess Poincare recurrence has no impact on a system that is essentially acausal. Random fields model correlations without a commitment to underlying causal models (but also without insisting that there cannot be underlying causal models). The kind of classicality that I've constructed, which in a sense is a half-way house, is not necessarily something that Einstein would have condoned.

Peter.

To do some riffing of my own:

I have been educating myself on your precise meaning of "continuous random field," and so read your NKS 2007 paper. Very nice. Now that I have a better understanding of the probabilistic structure of the field ("fractal structure all the way down") I think I have a better grasp of the dissipative mechanism though I would still want to argue that recurrence will obviate time asymmetry, for the reason that there are no boundary conditions on correlated causes from -oo to +oo. Yes, of course, I don't believe Einstein would approve of postulating continuous functions, even if probabilistic, without boundary conditions. I have to imagine that the system is closed, because correlation of causes implies endpoints which are correlated and time symmetry is preserved.

On the other hand, I appreciate that correlated observables in this random field model amount to a distributed cause, rather than a common cause between two particles--a large scale analog, I think, is laterally distributed control in complex systems science; e.g., Bar-Yam's Multiscale Variety. Then we could speak of self organized critical correlations, and correlations of correlations--I think I am starting to grok your emphasis on "contextual measurement" in modeling experiments; quantum averaging would thus subsume the measurement problem in a very natural way, would it not?

As they say in show business, I think this approach has legs.

Tom

Many thanks, Tom. Keep in mind that this is **just a model**. What happens "at infinity" cannot be experimentally tested. There's just how far we've managed to look so far. In this class of models there is a boundary condition that is largely the same at infinity as for QFT: the vacuum state looks the same wherever you are, no matter how far you go. That's enough to make the mathematics well enough defined to do honest business. We also can't measure the infinitesimally small, which this model includes information about, but it's all only as a place-marker, on the doubtless faulty basis that small-scale details are the same as at scales we /can/ examine, until we find ways to examine and understand Planck-scale Physics and beyond.

I'm not quite clear what you mean by imagining the system to be closed. It's just a model, closed or open is billions of light years away, and GR/QG issues presumably become important much sooner.

Yes, insofar as causes can be inferred from correlations, they are distributed.

Yes, contextuality --- in a field sense, which can be elaborated as a conceptually unexceptionable classical holism (I like to think that Bohm would like this aspect), not in a contextual particle property sense, which goes too much against the grain of classicality --- is essential. The relationship between classical measurement theory and quantum measurement theory cannot be understood without recourse to something like contextuality. If you haven't read "The straw man of quantum physics", it elaborates on how to think about measurement as a coarse-grained equilibrium of the field (at a fine-grained scale, there are discrete, non-equilibrium measurement events happening all the time, but the statistics of the measurement events have to be time-translation-invariant, for the events to be taken to be an ensemble; equilibrium, even if only coarse-grained, is a holistic property of the whole experimental apparatus, which feeds into Bohr's views on measurement nicely); straw man is very short!

By "correlations of correlations", do you mean n-point correlations, n>2?

I'm also not quite sure what you mean by "quantum averaging". I think the measurement problem is not a problem in a random field approach, although I have so far failed to elaborate the story at all well.

If this has legs, the first decent mathematician who comes along will leave me in the dust. But I've been waiting for it all to fall apart for all of the dozen years I've been trying to bring it together, so I'd be all too happy to have company on the road even if it's in front.

I appreciate that it is just a model, Peter. Is it, or can it be made, a unitary model? Then, being algebraic--or at least using algebraic tools--it presents the possibility of a closed form expression that mediates a scale-invariant measure that would obviate the notion that the basis for Planck scale measure is "doubtless faulty." We come back to Einstein's show-stopping observation that "From the quantum phenomena it appears to follow with certainty that a finite system of finite energy can be completely described by a finite set of numbers (quantum numbers). This does not seem to be in accordance with a continuum theory, and must lead to an attempt to find a purely algebraic theory for the description of reality. But nobody knows how to obtain the basis of such a theory." You might. You might even have already.

I should have been clearer about what I meant by closed system in this context--I meant, a scale invariant measure that guarantees self-similarity and self-limitation on the system from -oo to +oo. In other words, there is a certain context in which Planck's constant is zero and the world is smoothly classical, which would preserve your path toward reconciling classical and quantum field theories. Insofar as systems that are infinitely self-similar and self-limiting are by definition self-organized--time dependency is assured: the system evolves.

Given this condition of self-organized constraint, and distributed correlations of activity, I am persuaded that Nature's unifying principle is an information theory so far applied only to social and biological systems, that of Multiscale Variety (Bar-Yam) [2]. This theory seems to me sufficient to model discrete energy levels of fundamental physics in a continuous and self organized universe. Empirical results based on this theory show that time is the least significant measure of the state of the system [3] in that while states vary--and may radically vary--according to one's choice of time scale when observing, the aggregated time will show the system near equilibrium (a state of "dynamic centrality"). This corresponds exactly to your "...measurement as a coarse-grained equilibrium of the field..."

By "correlations," I meant only the correlated activities of a self-organized system.

Yes, I had read "straw man," which is why I was a little surprised that my context for "quantum averaging" was not clear. You use an averaging principle of measurement; the term itself is used by Brian Greene [4] to describe the effect of pixilated screens in forming a coherent image.

We are already on the same road, to understanding how an apparently continuous world expresses itself in discrete phenomena. Getting beyond the Planck barrier is essential to analytically smooth functions--"As the true lawfulness of nature, according to Leibniz's continuity principle, finds its expression in laws of nearby action, connecting only the values of physical quantities at space-time points in the immediate vicinity of one another, so the basic relations of geometry should concern only infinitely close adjacent points ('near geometry' as opposed to 'far geometry'). Only in the infinitely small may we expect to encounter the elementary and uniform laws, hence the world must be comprehended through its behavior in the infinitely small." [5] Taking the road analogy further, in my own point of view, the road may be lined on both sides by an impenetrable thicket, and we can see only the horizon forward and backward as we walk in a one dimensional line on a two dimensional surface. With access to the third dimension, rising above the road, we may find that the thicket is really exceedingly thin, and broad open meadows lie beyond. Kronecker is supposed to have said, "God created the natural numbers, all else is the work of man." I disagree. I think God created analysis, and the integers are the work of man. (I mean this metaphorically; I am not a believer.)

If nature is well ordered from hyperspace, n-dimensional analysis applies, and a continuous field theory is both extradimensional and time dependent.

[1]Einstein, The Meaning of Relativity, appendix II.

[2] necsi.edu/projects/yanee/NECSITechnicalReport2003-11.pdf

[3] necsi.edu/research/networks/com122/index.html

[4]Greene, The Fabric of the Cosmos.

[5]Herman Weyl, Philosophy of Mathematics and Natural Science.

I messed up the link in ref 2. It's

necsi.edu/projects/yaneer/NECSITechnicalReport2003-11.pdf

Tom

Tom, Thanks for the above. I'm sorry the perspective above doesn't seem to me to come through in your FQXi essay.

I think, without looking at it, that I understand your invocation of self-organization to be a way of understanding thermodynamic transitions of things like photographic emulsions and CCDs. For me, at this stage, these thermodynamic transitions are essentially given -- needing detailed explanation, but not getting it from me. Physicists take the "something" that causes the events to be "particles", even though they use the field formalism of quantum optics extensively and acclaim the empirical success of QED, etc. I've engaged with the field side of the relationship. I say that embedding thermodynamically nontrivial apparatus (stuff that's rather carefully engineered to be that way) in different fluctuating fields will result in different statistics of events.

On unitarity, a self-adjoint generator of time translation can be introduced consistently with the rest of the algebraic structure in the Lie field formalism (in section IV of "Lie fields revisited", particularly). Unitarity is shouted around the place so often, however, that I'm never sure that I've caught all the intended meanings. So is that enough, do you think?

I've never been very happy about scale invariance as a fundamental symmetry in Physics -- but then time-reversal invariance is about equally not part of our experience. We can require scale-invariance of the measurement algebra as a way of making scale-invariance breaking explicit in the state, I guess.

The one really hopeless aspect of talking to people is that sometimes you have to read stuff; in this case, your [2] and [3] above. My daughter Eleanor wants lunch, so bye.

Peter, my FQXI essay doesn't capture this POV, because I assume quantum mechanical unitarity (the wave function and its complex conjugate) as a condition of event length 1 probability (p.5); i.e., because my model is discrete, I can only speak in terms of the simplest field (0,1). This was never very satisfying to me, because locality demands a field result--at least renormalizable if not intrinsically normal--that allows quasi-classical smoothness without sacrificing "...different fluctuating fields (that) result in different statistics of events," as you say. Do you see what I am getting at?--I think I put it succinctly in my InterJournal archive article "On breaking the time barrier" (p. 4, 2.5): "We conjecture that just as we 3-dimensional creatures with 4-dimensional brain-minds arrive at such statistical results as central limit and regression to the mean by sampling large numbers of time-dependent events, Nature arrives at order by sampling large numbers of hyperspatial events that we interpret as the flow of time." It doesn't matter that your model is not extradimensional--(I have shown in my FQXI paper that the 4-dimension horizon is identical to the 10-dimension boundary)--if your QFT formalism can predict experimental results that vary according to the time scale chosen while holding field conditions constant (picture a guitar string of constant length resonating with a string plucked at successively different lengths, which we can liken to successive moments), then string field theory can well explain the violation of time reversal invariance. Here's why:

When you deny locality (particle properties) to a measurement, in favor of your resonance effect, you automatically introduce the positive energy operator and thus the temporal direction. I.e., all future information is contained everywhere on the constant length "guitar string."

You have to pay the price of scale invariance, however, for time asymmetry. I mean, that unless one obviates the time barrier of the Planck scale (Planck's constant = 0), classical determinism is as time reversal invariant as it ever was. Scale invariance guarantees that quantum mechanics is coherent to an infinity of unitary observables (again, I mean the wave function and its complex conjugate). But who says that t=0 must be a singularity?--suppose there are an infinity of zeros and every one is normal. Then the simplest field (0,1) is both continuous and random.

Chaitin's constant Omega is such a normal number--one can never predict the next digit of Omega in a calculation, and so the value of this number is "maximally unknowable," as Chaitin puts it. Just because the discrete digits are unknowable in the present, however, does not imply that strings of random digits can't resonate (correlate) to a specific value in the future. In fact, they have to. Do you see further what I am getting at? -resonating values on a pair of strings correlate observable effects without necessitating point particle properties (locality, e.g.). With apologies for quoting myself again, in my ICCS2007 paper, I wrote (p.4, 2.2.2) "...we conjecture that if the choice of (program) determines the outcome of a present computation--the future result resting in a future computer maximally unknowable in the present--then the result _exists_ in a context of information-richness, and what we know of the result is information-poor. In other words, an infinite number of future computing machines calculating an infinite number of results based on the same algorithm for computing Omega, gives us for every finite computing machine in the present a unique result of infinite complexity that is self-similar to the infinite set of all results on an infinite number of future computing machines. The aggregated result is infinitely self-similar, in other words."

I can't see that the algorithmically random Omega differs from a random continuous field ("...fractal all the way down.") It corresponds to your system of "...different fluctuating fields...different statistics of events."

Therefore, the resonant state of your measurement apparatus, with the measurement value, is time dependent and unitary, independent of scale--meaning that classically smooth functions are available "all the way down," so long as one does not insist on a singularity (the "guitar string" is continuous).

Thank you for indulging me this dialogue, Peter.

Please give a second thought to the role of scale invariance.

Tom

Tom, I'm sorry to say that with your last I think we are, for now, approaching things too differently. I'm intending to work in a conceptual environment that is recognizable to a moderately conventional but open-minded Physicist, whereas the appearance to me is that you are too engaged with complex-systems ways of thinking for me to make more than superficial contact with you.

Please note that I make no claims for ultimate superiority of my approach; as far as I'm concerned it's just an accident of my intellectual history. I would say that a convergence or reconciliation of quantum/random field models and complex-systems modeling may well be desirable, but /I/ can't see how it would come about.

There are hints in what you say about scale invariance of what might be interesting mathematics, but I can't at present see how to do the mathematics myself. In my terms, I would like to see an explicit presentation of an algebra of observables that is scale-invariant, and a discussion of the differences, similarities, and relationships between a Lorentz/Poincare invariant quantum/random field and a scale-invariant quantum/random field. The trouble is perhaps that I've been thinking, haltingly, in terms of algebras of observables for long enough that I am now relatively deaf to ways of discussing traditional, very-poorly-defined Hamiltonian and Lagrangian formalisms for quantum theory.

You mention positivity of energy. I have some things to say about positivity of energy as a result of an e-mail correspondence with someone else that I will post separately, because their comments crystallized well.

Someone has been kind enough to point out to me in the last few days, and politely, which was nice, that "Of course one has to impose the positivity of the energy", and that anti-particles "are just free particle solutions running backward in time". I think the mathematics of my FQXi paper itself is intrinsically relatively indifferent to these questions, but they are important questions for anyone who is making a transition from only thinking in conventional ways about QFT. Positivity of energy is a Wightman axiom and is as important in the Haag-Kastler axioms, which are only two visible tips of its appearance in Physics. I had the following three responses:

1) The Lie random field formalism is a block-world formalism. There is no question of "stability" in the conventional sense; adding a positive or a negative frequency component to a model just changes the results of measurement in different ways. There is a block-world sense in which there is no evolution of a state, even though a unitary translation operator can be constructed, because a state fixes what the results of measurements would be for all time. Note (again; I'm a stuck record) that for me this is not an ontological claim about the Nature of Time, only a description of the mathematical components of this kind of model. Indeed, I regard this kind of model as too grounded in correlations without reference to causality to be entirely satisfactory. I'm offering an intermediate form that makes us able to model and think about Particle Physics in two ways that are different enough to move us forward in future, even though the confusion of having two very different ways of modeling and thinking will at first be counterproductive.

2) Negative frequencies are not so bad. Consider that a random field is, in a sense, a classical system. If we were talking about a classical continuous free field, positive and negative frequencies would both be positive energy, both different from the zero field. If we further consider a classical equilibrium state of a classical field, the equilibrium state is not the minimum energy state of the field (which is the zero field), but subtracting a mode from the field increases the free energy of the field, even though we decrease the classical energy. The equilibrium state is /thermodynamically/ stable. Adding or subtracting both negative or positive frequency modes will all have a free energy cost. There are distinct concepts of energy. Note that the vacuum state, being Lorentz/Poincare invariant, is more invariant than an equilibrium state, which is only invariant under the little group of a time-like 4-vector.

3) A free theory is CPT-invariant, but time-reversal changes particles into negative-frequency anti-particles. QFT has conventionally constructed an algebraic structure that ensures that anti-particles have positive energy. If we do so, we end up with a conventional QFT, in which we cannot, apparently, rigorously construct an interacting theory in Minkowski space. If we introduce negative frequency fields without trying to make them positive energy -- largely ignoring energy and action because they are nonlocal, unmeasurable secondary concepts for a random field, and focusing instead on an algebraic way to generate observable statistics and correlations -- then we can construct rigorous interacting Lie random field models. That seems like a trade-off we might want or be willing to make. The very existence of negative-frequency anti-particle fields perhaps even suggests that we should go this way.

Energy and action arguments are a fundamental part of informal discussion of quantum field theory, so they cannot be too casually set aside (despite energy and action being rather ill-defined -- for an interacting field, using third and fourth powers of a distribution, rather horribly so). There's no reason to think, however, that energy arguments should be as effective in a mathematically well-defined block-world formalism. I hope that losing some aspects of energy and action is not such an enormous conceptual change that it puts Lie random fields into the distant future or not-at-all for Physics, but we will see. A lot is different, perhaps too much, but something that lets us more-or-less understand quantum (field) theory after 80 years of effort is unlikely to be easy.

The Wightman axioms and the Haag-Kastler axioms overconstrain algebras of observables, except for free fields. We could lose other axioms to allow us to construct interacting theories, but positivity of energy seems a natural candidate in a block-world formalism, given that such a move allows us to construct interacting Lie random fields.

My correspondent's formulation of the question was particularly clear, polite, and Physics-y. Thanks to them.

Okay, then. I wish you success, Peter.

Tom

I point out, for anyone coming by here, that the current issue of "Studies in History and Philosophy of Science Part B: Studies in History and Philosophy of Modern Physics", Volume 39, Issue 4, Pages 705-916 (November 2008), is a Focus Issue on "Time-Symmetric Approaches to Quantum Mechanics". A number of articles that anyone thinking about the direction of time might expect to read.

Peter,

Great essay. I especially appreciate your statement about Hermiticity:

"In this Hilbert space formalism, in other words, the choice of a direction of time is the difference between classical and quantum fields property so commonly used in quantum mechanics."

It is unfortunate, however, that most have not fully appreciated the physical implications of complex analysis. We toss out reverse time and conjugate solutions when we are faced with a perplexing imaginary term. Indeed, in the view of unitarity- the new form of conservation under the Hilbert formalism- we finally see the universal truth through quantum mechanics. I hope you will read my essay too, where I reveal that our entropic viewpoint has blinded us from the anti-reality of time reversal, charge conjugation, and the truth of a harmonic universe.

Furthermore, I might suggest: perhaps it is not that time does not exist- it is, rather, that time maintains zero expectation value- just like everything else. In an instant, when we precipitate a quantum mechanical transaction in an expectation value integral, time may not exist. However, if we choose a different basis, as we may by a unitary transformation, we will see time "ebb and flow" periodically, to varying degree.

Hi Ryan, thanks for your comment. I've now commented on your essay on its comment stream. I often think that complex analysis is purely there because of the utility of algebraic completeness, and the notation of Fourier analysis is much easier if we do cos and sin in a unified way. If that means we have too many DoFs, however, there will presumably be better and worse ways to constrain the mathematics so that our experimental data is sufficient to determine parameters in our theory. Restricting to positive frequency works pretty well for predicting empirical data, but I consider that whether it's the most effective way should be up for discussion.

I believe that unitarity is not a good thing for everyone to be so obsessed about. It's enough for a presentation of an algebraic structure to be Lorentz/Poincare covariant. There are other important algebraic structures, for example associativity of a *-algebra of observables and positivity of a state over the algebra, which are enough to allow a probability interpretation (chapter III of Haag's "Local Quantum Fields" is the best way to this, but it's still relatively heavy math). An elementary paper that set my mind pretty much at rest over complex numbers and has been a motivating force for well over ten years even though I doubt I will never be able to cite it formally, is Leon Cohen, Foundations of Physics 18, 983(1988), "Rules of probability in quantum mechanics" --- a Hilbert space and operators are a mathematically effective way to generate probability densities, expected values, and correlations.

I'm afraid that I don't understand the last paragraph of your comment, either from my POV or as a statement in the conventional ways of talking about QM. I'm sorry not to be able to give you even a paraphrase of what I think it says to help you reformulate it.

Thanks for elaborating on your approach.

I do agree with what you say! The point at which we differ, I think, is with reality vs. completeness. You refer to taking the positive frequencies only, and to Lorentz/Poincare covariance. Perhaps they are sufficient to describe what we observe, but a complete theory might also acknowledge things which we cannot observe directly (after all, we know there exist dark matter and dark energy). In CPT symmetry, how will you account for antiparticles without allowing negative frequencies. Even if we don't always use them in our day-to-day computations- they are there in the mathematics. Thus, the symmetry is important. If I face the wall of my bathroom and look into the mirror, I can now see myself. Furthermore, I can even see out the window: wonderful things may lie in the opposite direction.

Regarding your last paragraph about my last paragraph ;-)... I did not clarify the difference between time-independent and time-dependent. What if we let something other than time be our "external" parameter in Schrodinger's equation? I meant to say that we may change our basis so we don't "see" time. "Exist" is a stretch, but I do believe existence is revealed by observation.