When people write down a list of axioms for quantum mechanics, they typically neglect to include a crucial one: positivity (or more generally, boundedness below) of the energy. This is equivalent to saying that something very different happens when you Fourier transform with respect to time versus with respect to space. If $\psi(t,x)$ is a wavefunction depending on time and space, and you Fourier transform with respect to both time and space

$$\widetilde{\psi}(E,p)=\frac{1}{2\pi}\int_{-\infty}^\infty \int_{-\infty}^\infty \psi(t,x)e^{iEt}e^{-ipx}dtdx$$

(the difference in sign for $E$ and $p$ is just a convention) a basic axiom of the theory is that, while $\widetilde{\psi}(E,p)$ can be non-zero for all values of $p$, it must be zero for negative values of $E$.

This fundamental asymmetry in the theory also becomes very apparent if you want to “Wick rotate” the theory. This involves formulating the theory for complex time and exploiting holomorphicity in the time variable. One way to do this is to inverse Fourier transform $\widetilde{\psi}(E,p)$ in $E$, using a complex variable $z=t+i\tau$:

$$\widehat{\psi}(z,p)=\frac{1}{\sqrt{2\pi}}\int_{-\infty}^\infty \widetilde{\psi}(E,p)e^{-iEz} dE$$

The exponential term in the integral will be

$$e^{-iE(t+i\tau)}=e^{-iEt}e^{E\tau}$$

which (since $E$ is non-negative) will only have good behavior for $\tau <0$, i.e. in the lower-half $z$-plane. Thinking of Wick rotation as involving analytic continuation of wave-functions from $z=t$ to $z=t+i\tau$, this will only work for $\tau <0$: there is a fundamental asymmetry in the theory for (imaginary) time.

If you decide to define a quantum theory starting with imaginary time and Wick rotating (analytically continuing) back to real, physical time at the end of a calculation, then you need to build in $\tau$ asymmetry from the beginning. One way this shows up in any formalism for doing this is in the necessity of introducing a $\tau$-reflection operation into the definition of physical states, with the Osterwalder-Schrader positivity condition then needed in order to ensure unitarity of the theory.

Why does one want to formulate the theory in imaginary time anyway? A standard answer to this question is that path integrals don’t actually make any sense in real time, but in imaginary time often become perfectly well-defined objects that can be thought of as expectation values in a statistical mechanical system. For a somewhat different answer, note that even for the simplest free particle theory, when you start calculating things like propagators you immediately run into integrals that involve integrating a function with a pole, for instance integrating over $E$ integrals with a term

$$\frac{1}{E-\frac{p^2}{2m}}$$

Every quantum mechanics and quantum field theory textbook has a discussion of what to do to make sense of such calculations, by defining the integral involved as a specific limit. The imaginary time formalism has the advantage of being based on integrals that are well-defined, with the ambiguities showing up only when one analytically continues to real time. Whether or not you use imaginary time methods, the real time objects getting computed are inherently not functions, but boundary values of holomorphic functions, defined of necessity as limits as one approaches the real axis.

A mathematical formalism for handling such objects is the theory of hyperfunctions. I’ve started writing up some notes about this, see here. As I find time, these should get significantly expanded.

One reason I’ve been interested in this is that I’ve never found a convincing explanation of how to deal with Euclidean spinor fields. Stay tuned, soon I’ll write something here about some ideas that come from thinking about that problem.

Isn’t it possible to create some ‘negative energy’ with the Casimir force?

Pascal,

Energy is basically only defined up to a constant. You can always shift where E=0 is, so the point really is that energies are bounded below, that they can go off to infinity in only one direction. That’s the asymmetry in the theory.

Of course, in classical GR there is an absolute zero of energy density, corresponding to flat spacetime, right?

Douglas Natelson,

Yes, in classical GR. But I’m just trying to make a simple very general point about quantum theory, where very generally you have a Hamiltonian and it generates time translations. The positivity (perhaps after a shift in definition) of the energy eigenvalues is a very general fundamental aspect of a quantum theory, and if you work with Euclidean QM or QFT, this gets reflected in an asymmetry in the behavior in imaginary time. If you think about Euclidean QFT this is kind of surprising, since the theories one works with are often formulated in a Euclidean invariant manner (e.g. as a Euclidean invariant path integral or stat mech system).

there’s a refection in your text that I believe should be a reflection

There are two instances of “the the theory” in the post.

Sabine and Aula,

Thanks! Fixed.

The relation between the Euclidean and Lorentzian theories is most transparent for specific set of quantities, which are correlators of time-ordered operators evaluated between the vacua in the asymptotic past and future (which may be different for a time-dependent Hamiltonian). This is great if you are interested in S-matrix elements, which is typically the case if you are a particle physicist.

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But these are not the most general quantities you may be interested in, for example you may be interested in expectation values of Schrodinger picture operators in some excited state (as in measuring the CMB), or in some retarded correlators corresponding to physical measurements in CM physics. For these quantities, the Euclidean calculations are less useful. In fact, it is not clear to me if the Euclidean calculations alone contain all the information needed in principle to obtain the Lorentzian information (I’d be interested in opinions), but at the very least extracting that information is somewhere between difficult to impossible (for example the analytic continuations involved often do not commute with perturbative expansions).

Moshe,

Thanks. I very much agree that the relation between Minkowski and Euclidean calculations is non-trivial and deserves much more attention. The usual assumption, based on the kinds of S-matrix calculations you mention, seems to be that all that’s involved is changing some factors of i, but there’s much more to it.

Moshe, Peter: There certainly is much more to Lorentzian Euclidean relation than changing factors of “i”. For example, blindly applying this to an arbitrary metric gives a complex metric field that has no natural interpretation. Besides, Euclidean quantities (such as Greens functions) do not distinguish between coincidence and null limits, while null surfaces play a crucial role in Lorentzian regime. However, there does exist a covariant alternative to Wick rotation which do not have these difficulties, though much less work has been done on it.

DKepler: if not off topic, could you say more about this covariant alternative to Wick rotation you are taking about ?

martibal,

(Peter can decide if the question/reply are off-topic.)

The key idea behind this (first noted, to the best of my knowledge, by Hawking & Ellis (HE)) is based on the following mathematical result: given a Lorentzian metric g_L and a nowhere vanishing timelike direction field U, one can always construct a Euclidean metric g_E. As is well known, non-compact manifolds always admit such a vector field, as well as compact manifolds with Euler number zero.

Some references where this HE observation was discussed in the context of QFT are Candelas & Raine, PRD15, 1494 (1977) and Visser 1702.05572, while recent generalization and consequences for GR, euclidean QG, and euclidean action can be found in Kothawala, arXiv:1705.02504, arXiv:1802.07055. The latter references also discuss transition from euclidean to lorentzian, instead of just the euclidean phase.

Hope this helps.

DKepler: thanks a lot, this helps a lot.

There is also a connection between positivity of energy and causality. If one takes a derivation of the relativistic electromagnetic vector potential from a four-current (e.g. here), and requires positivity of photon energy as a supplement to the Maxwell equations, then the retarded solution is the only possible.

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Dear All,

Just recently I happened to come across a related problem. I would be interested in some comments on the below, related phenomenon. (Peter might decide if it is off-topic.)

In QFT, in a Lorentz signature setting, one can define two kind of quantum field correlator function(al)s: simple VEV of products of the quantum field operator (let us call them Wightman correlators), and the VEV of time ordered products of the quantum field operator (let us call them Feynman correlators). The Wightman correlators are the ones which satisfy the Wightman axioms, including the Wightman positivity. (If Wick rotated, they will satisfy the Osterwalder-Schraeder axioms, including the OS positivity, or reflection positivity.) The Feynman correlators, however, are the ones which are in principle returned by a Feynman integral procedure. Also, these are the ones, which turn up when one would like to evaluate QFT predictions (S matrix, for instance). To me, it seems that the transition from Wightman correlators to Feynman correlators is basically a projection (time ordering). So, if I am not mistaken, Wightman correlators (without bringing in some external information) cannot be fully recovered from merely the Feynman correlators. The question naturally arises: to what extent the Wightman axioms (in particular, the Wightman positivity) are reflected in the properties of the Feynman correlators? (We should not assume, of course, anything to be known about the theory, except for its Feynman correlators.)

Best regards,

Andras

Andras Laszlo,

In practice, it is not trivial to obtain the Wightman functions from T-ordered correlation functions (or their connected versions, which can also be defined. For example the connected T-ordered 2-point function is the retarded commutator of the two field operators). As you say, going the other way can be done straightforwardly.

In principle, however, I THINK it can be done. The T-ordered Green’s functions fix the vacuum state. Then the Wightman functions are vacuum expectation values of field operators.

Thinking on the matter a little more, it HAS been done where constructive field theory methods work (in for example $\phi^{4}_{d}$ models for $d<4$). In these models

the time-ordered functions are obtained (in Euclidean space, where they are analytic continuations of those in Minkowski space), and the Wightman axioms have been proved. I'm not an expert on continuum constructive field theory, but this is an example.

Dear Peter Orland,

Thanks, and could you give some hints where one could start looking at these claims?

(What I actually wanted to ask: what one can say _without_ knowing the field operators — i.e. without actually solving the QFT model etc. That is, how the Wightman positivity translates to a situation if only the Feynman correlators — i.e., time ordered correlators — are known about the model.)

Best, Andras

Andras,

I’m not going to satisfactorily answer your questions, but I’ll make some remarks, which I hope generate more light than heat.

As to your first question, the book by Glimm and Jaffe is probably where you should start. I’m no expert, so I’ll tell you what little I know (I’m very familiar with these concepts on a lattice, but in the continuum, don’t trust my statements as the last word). In constructive field theory, it is important to have reflection positivity, which is a statement about observables on a half space ${\mathbb R}_+\times {\mathbb R}^{d-1}$, which is in Euclidean space. This is a sufficient (but perhaps not necessary) condition for unitarity and the spectrum condition. In constructive field theory, the models are not actually solved, but bounds are place on the convergence of resummation of perturbation theory (by a Borel transform). Once the Euclidean axioms are proved, then the claim is that, after a Wick rotation, the Wightman axioms are satisfied.

Now in a cut-off theory (requiring renormalization, which is very different from the axiomatic approaches), in principle, the time-ordered correlations define the vacuum state. If you knew the vacuum, you can certainly construct these, but the inverse procedure should be possible to find the vacuum from them. For example, in a free field theory, the two-point function determines a unique Gaussian vacuum wave functional. I have not thought more deeply as to how to do this more generally. It sounds like a good project for a student, though.

Anyway, once you have the vacuum state, the un-time-ordered correlations can be found. Of course, nobody actually knows the exact vacuum state for most interacting field theories, so I am stating a matter of principle.

One more remark. Given the time-ordered functions, the connected Green’s functions,

can be found. These are (as I said in my previous statement) a little different. For example, the two-point connected function is the vacuum expectation value of the retarded commutator. The advanced commutator of two fields can also have its expectation value found this way. So at least vevs of commutators can be found. This is close to the two-point Wightman function, but perhaps not good enough an answer to you question.

Dear Peter Orland,

Thank you for the answers.

Best regards, Andras

I am not sure if the has to be bounded from below in quantum mechanics. If it is and the Hamiltonian is only defined as a quadratic form than the Friedrichs extension guarantees the existence of a self-adjoint operator but if you had a s.a. H from another source, my intuition would be that you often can avoid running into problems even if it is unbounded from below. Take for example H= sqrt(p^2+m^2) + Z/r a toy model for a relativistic hydrogen atom. If Z is big enough, this is unbounded from below. But please correct me if I am wrong.

In QFT, however, I agree that you want a spectrum condition. My preferred reason is that even in the absence of a preferred vacuum (for example in the absence of Poincare symmetry because auf background fields) it allows to define composite operators like phi^2(x) via normal ordering (subtracting the dev of that expression in your favourite vacuum) as you need it for example for an energy momentum tensor: If both your reference vacuum state and your current state obey a positive spectrum condition, you get smooth expectation values of your composite operators. This is for example (in the example of QFT on curved manifolds) described around Thms 6.3 and 6.4 in Fewster’s lecture notes https://wiki.physik.uni-muenchen.de/TMP/images/9/9f/Lecturenote-3908.pdf