Observables are expressed by Hermitian operators in quantum mechanics; their eigenvalues are real. In other words, the eigenvalues lie at a special curve in the complex plane – namely the real axis. Of course, it isn't the only curve that may be preferred. For example, unitary operators obeying \(UU^\dagger=1\) have eigenvalues whose absolute values are equal to one, i.e. \(\exp(i\phi)\). A place on a twotorus could naturally be described by two unitary operators, a description wellknown to the people familiar with membranes in Matrix Theory.
But many things in physics are naturally complex. For example, we routinely extrapolate the values of solutions to complex values of the time (thermal calculations) or complex values of energy and momenta (in the discussions about analyticity). In general, fields are functions of coordinates or momenta so for complex values of these coordinates and/or momenta, they take on complex values, too.
We still like to use specific theories with specific reality conditions – according to some specific choices that just "work". But are these reality conditions necessary? Could they be different? If you imagine a beautiful analytic structure behind string theory, could there be some deeper principles that tell us which "real sections" of the complexified configuration spaces and moduli spaces are the real ones? Is there any freedom?
Let's return to quantum field theory. Consider a real scalar field \(H\) whose potential is
\[ V(H) = (H^2+1)^2. \] For real values of \(H\), this is a nonnegative function. On the real axis, it only has one stationary point, the global minimum at \(H=0\). However, if you allow complex values of \(H\), you find other special points, too. The derivative of the potenial is
\[ V'(H) = 2H (H^2+1) \] and vanishes at three points. Aside from \(H=0\), there are two extra stationary points away from the real axis, namely at \(H=\pm i\). Do they play any role in physics?
In the discussions of analyticity, singularities of scattering amplitudes and similar functions of the momenta do play a role even if they occur at unphysical, complex values of the momenta and energy. Such poles of the scattering amplitudes may describe stationary states, bound states appearing as resonances, and so on. Note that in quantum field theory, such singularities arise from propagators whose poles are located away from the real axis if the corresponding particle becomes unstable, for example.
But what about the stationary points of the potential away from the real axis? I've been asking similar questions often when trying to identify some preferred pure or mixed states in string theory that serve as initial conditions of the Universe.
A funny aspect of the potential above is that the potential is actually guaranteed to be real even if \(H\) takes values that are not real. In particular, if \(H=ih\) is pure imaginary, the potential changes to something we know very well:
\[ V(h) = (h^2+1)^2 \] This is, of course, the usual Higgs potential that spontaneously breaks various symmetries and is used in the Higgs mechanism. Can a single theory allow you to jump from the real axis to the pure imaginary axis and/or vice versa? And if it is not allowed, what's the actual reason?
In the discussion above, I implicitly required that the potential has to be real. This requirement is a part of the answer to the question in the previous paragraph. It is needed because it's a part of the Hamiltonian that should be Hermitian for the evolution operators to be unitary i.e. for the total probability of all mutually excluding alternatives to be conserved. But the reality of \(V\) doesn't imply that \(H\) had to be real.
Instead, \(V\) is real at 4 curves: the real axis, the pure imaginary axis, and two pieces of a hyperbola going through the \(H=\pm i\) points. The hyperbola is inappropriate as a good range where we might allow \(H\) to vary. Why? Because the natural kinetic energy density wouldn't be real, unless we defined it as the real part of some expression.
However, the real and imaginary axes are OK for \(H\): the potential energy density and the kinetic energy density end up real in both cases. I deliberately started with the relative plus sign in the potential in order to make you (and me) think that the normal Higgs potential doesn't have to be the "default option"; it could be a complexification of a different, more primordial "real axis" which was allowed sometime at the beginning of the Universe.
The idea is that \(H=0\) is a minimum of the original potential, something that isn't true for the Higgs symmetrybreaking potential. It's intriguing to imagine that the Universe could have been sitting at this stable point before it decided to switch to the symmetrybreaking values \(H=\pm i\).
Now, an important technical question is whether the transition from a real \(H\) to a pure imaginary one could have proceeded continuously by turning by 90 degrees at zero; or whether one needs some tunneling. The first option may look strange. Classically, the differential equations will always evolve a real \(H\) to another real \(H\). That's why we assume the same reality conditions in quantum mechanics but is it really necessary that quantum mechanics forbids the transition from the real axis to the pure imaginary axis?
And if it does ban such a continuous transition, rolling from the minimum at \(H=0\) to even lower values of the potential (which is possible because \(H\) becomes pure imaginary), can't Nature allow tunneling from a real \(H\) to \(H=\pm i\)? If such events were possible, it could naturally explain why our Universe seems to be pretty close to various points of instability.
This is how we may describe various paradoxical hierarchies we observe in Nature.
The Higgs vev is famously very small in the Planck units, something like \(10^{15}\). If the reasoning above has anything to do with Nature's tastest, the smallness could have something to do with Nature's being at the \(H=0\) point in the past. This point is unstable if we demand \(H\) to be pure imaginary i.e. \(h\) to be real. However, it is stable as a function of a real \(H\).
Nature may have been at such points – which seem as unstable, saddle points today – in the past and it could have found convenient other places to tunnel to. The nearby stationary points in the complexified (yet complicated) landscape would have been preferred because shorter tunneling events have a higher (faster) rate.
Such an explanation could also be helpful for inflation (assuming, for a while, that the inflaton is something different than the electroweak Higgs: there actually exist models in which the Higgs field we now know well also acts as the inflaton as long as it has some nontrivial and strong coupling to the spacetime curvature.
Finally, this setup could even have the potential to explain the nearboundstate of the dineutron or its isospin partner, the \(J=1\) deuteron. There are no particles like that but the two nucleons only fail to be bound by 60 keV, a tiny amount (3+ orders of magnitude too small) relatively to the hundreds of MeV expected in generic nuclear situation. This was a bizarre case of finetuning that Nima ArkaniHamed mentioned in some recent talks. This smallness of the 60 keV figure may be an accident; it may have an anthropic explanation we don't know; it may have an oldfashioned explanation based on some expansion that makes the smallness manifest and at least Nima doesn't know this expansion; or it may have something to do with tunneling and similar processes in the early Universe.
Of course, these intriguing possibilities that remain vague are just small components of my plans to determine the preferred initial state of the Universe according to string theory if there's any. Many things we can't explain could have explanations in terms of some processes affecting the young Universe in which the complexified spacetime and perhaps the complexified configuration space is essential.
There's one subtle general issue I want to mention. Quite generally, we view the anthropic constraints to be independent of the oldfashioned physical constraints coming from impersonal mechanisms. Nature evolves in some way and may produce a large multiverse, the anthropic folks argue, and on top of that, one simply has to do the selection and prefer the Universe with conscious observers (or many such observers). Clearly, the two stages of the selection are independent: the anthropic selection couldn't have been explained by the oldfashioned physical processes. That's the very reason why physicists decided to think about the anthropic selection in the first place.
But Nature could actually be equipped with some physical mechanisms that are "useful for life" because of their intrinsic mathematical properties. For example, the existence of various large numbers and hierarchies seems to be needed or helpful for the evolution of life. For example, we need stars that are large and live for a long time (a condition for evolution of species to proceed). The lifetime of an average star depends on some of the dimensionless parameters that quantify the hierarchies.
Nature could actually have builtin mechanisms that prefer the evolution into vacuum states described by small dimensionless numbers (and related large numbers, like the number of nuclei in a typical star). The anthropic picture is that Nature mostly produces mess, googols of worthless Universes etc., and only when this stuff exists, we must choose the viable products out of the pile of mess. However, Nature could systematically direct the Universe towards the states that look like those that are hospitable to life. The fundamental laws underlying Nature could include stimuli for life. Hierarchies and nearby unstable points could be some of the stimuli.
And that's a speculative memo.
Thursday, March 08, 2012 ... /////
Reality of complexified fields
Posted by
Luboš Motl
at
11:24 AM



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snail feedback (6) :
Dear Mr. Motl,
The LateX equation objects imbedded in your site used to be resolved into pretty equations within a few seconds by my browser on my old laptop. But now I have a new machine running Windows 7; I use Windows Explorer. What do I need to install to be able to automatically see your equations come up as intended instead of raw LaTeX syntax?
Thanks,
Andrew
Dear Andrew, thanks for your report. Unfortunately, the reasons for a faulty behavior are too numerous. I don't know.
Try to install a different web browsers, at getfirefox.com or chrome.google.com, for example... BTW yours is called Microsoft Internet Explorer, not Windows Explorer. The latter only browses local files on your PC.
Me too, I see Latex stuff, not formulas. I have Firefox 10.
I won't be solving your computer glitches for free, OK? It views OK on all my computers and browsers, most of the time at least.
If you think that the problems isn't purely on your side, go to mathjax.org and complain there.
I've got convinced that I should use LaTeX on this site and MathJax is the best solution that exists and nothing can be improved about it. If you can't make it work, it's too bad but it's your fault.
Dear Mr. Motl,
Thanks for the advice. I downloaded Firefox and now the LaTeX output is being displayed after a couple of seconds.
(Yes, I meant Internet Explorer, not Windows Explorer. Oops.)
Regards,
Andrew
I use Firefox, and it works fine.
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