B Is imaginary time a fourth dimension?

[bobrubino]

In a conversation Dr. Stephen Hawking said that he used Imaginary time as a 4th dimension to show that there was nothing before the big bang. How is it possible for Imaginary Time to act as a fourth dimension when it is still part of the ordinary time dimension?

 

[Ibix]

Imaginary time is a fairly unhelpful concept, and why Hawking used it in his popularisations I don't understand. It's just time.

The point is that Pythagoras' theorem in three Euclidean dimensions is $\Delta l^2=\Delta x^2+\Delta y^2+\Delta z^2$. In four dimensions in Minkowski spacetime, however, the equivalent expression is $\Delta s^2=\Delta x^2+\Delta y^2+\Delta z^2-c^2\Delta t^2$, which produces a "distance" between two events in spacetime rather than a distance between two points in space. The minus sign encouraged some people to multiply the $\Delta t$ by $i$, the square root of minus 1, to turn the minus into a plus. Thus they have an "imaginary time" that is just the time coordinate everybody uses multiplied by $i$ and have concealed the minus sign.

It isn't a particularly helpful thing to do, IMO, and stems from a time when there seemed to be an effort to brutalise relativistic mathematics so it looked like non-relativistic maths. It's largely fallen out of favour now, but is worth knowing about because you occasionally come across it.

 

[bobrubino]

Thank you.

 

[haushofer]

In static spacetimes the i disappears from the spacetime interval if you use imaginary time, but for e.g. rotating masses (Kerr solutions) the i appears explicitly in the interval. That makes it hard to interpret.

 

[vanhees71]

Imaginary time is a fairly unhelpful concept, and why Hawking used it in his popularisations I don't understand. It's just time.

The point is that Pythagoras' theorem in three Euclidean dimensions is $\Delta l^2=\Delta x^2+\Delta y^2+\Delta z^2$. In four dimensions in Minkowski spacetime, however, the equivalent expression is $\Delta s^2=\Delta x^2+\Delta y^2+\Delta z^2-c^2\Delta t^2$, which produces a "distance" between two events in spacetime rather than a distance between two points in space. The minus sign encouraged some people to multiply the $\Delta t$ by $i$, the square root of minus 1, to turn the minus into a plus. Thus they have an "imaginary time" that is just the time coordinate everybody uses multiplied by $i$ and have concealed the minus sign.

It isn't a particularly helpful thing to do, IMO, and stems from a time when there seemed to be an effort to brutalise relativistic mathematics so it looked like non-relativistic maths. It's largely fallen out of favour now, but is worth knowing about because you occasionally come across it.

This is this unfortunate usage of the $\mathrm{i}c t$ convention which seems to go back to Minkowski himself.

Nowadays one works with Minkowski space as a real affine space with indefinite fundamental form of signature (1,3). Then everything is real, and $t$ or rather $x^0=c t$.

Imaginary time comes into use in contemporary physics only in a formal way when you want to treat many-body systems in thermal equilibrium. Then inverse temperature (or $1/(k_{\text{B}}T)$ in SI units) can be treated formally as an imaginary time, because the (grand-)canonical statistical operator $\propto \exp(-\beta \hat{H})$ is just formally like the time-evolution operaor $\exp(-\mathrm{i} \hat{H} t)$. So you can make $t=-\mathrm{i} \beta$.

In (non-relativistic or relativistic) field theory you can build up an "imaginary-time formalism" of perturbation theory. Then finite temperature can be treated as evolution along imaginary time from $t=0$ to $t=-\mathrm{i} \beta$ but with periodic boundary condtions for bosonic and antiperiodic boundary conditsions for fermionic fields/particles. This is known as the Matsubara formalism of manybody equilibrium thermal field theory.

 

[berkeman]

In a conversation Dr. Stephen Hawking said

Do you have a link to that conversation? It's always best to post links to references when you start a thread in the technical PF forums. Thanks.

 

[PeterDonis]

he used Imaginary time as a 4th dimension to show that there was nothing before the big bang

What Hawking is referring to, I believe, is his "no boundary" proposal for the beginning of the universe. This was just a proposal, and AFAIK there is no way to test it now or in the foreseeable future, so it's just a hypothesis, not established fact or even established theory.

 

[vanhees71]

Maybe it's a good idea to look for scientific papers about that idea. I only know the enigmatic statements in "A brief history of time", which imho is a highly overrated popular-science textbook.

 

[Algr]

Wouldn't imaginary time be a fifth dimension? The idea of multiple dimensions of time sounds interesting to me, but I I don't know what to call it. Is that what is meant by imaginary time?

 

[vanhees71]

No. In a certain sense you can define a mathematical scheme for quantum-many body QFT, where time is treated as a complex variable. That doesn't mean that there's a fifth dimension somewhere.

 

[Hill]

This is what MTW said about the imaginary time in Gravitation:

 

[vanhees71]

In addition there's another deeper reason for the specific structure of Minkowski space as a affine manifold with a fundamental form of signature (1,3) or equivalently (3,1).

You can derive the Lorentz transformation by just assuming the invariance of physics under change of the inertial reference frame + the assumption that any inertial observer considers "his space" as a 3D Euclidean affine manifold (with all it's symmetries, i.e., translations and rotations, forming the group ISO(3)).

Then an analysis of the possible transformation laws leads to three possible structures for the homogeneous transformations (i.e., those without translations, generated by the boosts and rotations):

(a) SO(4) -> Euclidean Spacetime
(b) Galilei group -> Galilei-Newtonian spacetime (a fiber bundle)
(c) $\mathrm{SO}(1,3)^{\uparrow}$ -> Minkowski spacetime

Now in addition one of the most fundamental properties any spacetime model must have is to allow to define a "causal structure", i.e., the "causal direction of time". This excludes (a), because in this case noting fixes in any way the temporal order for all inertial reference frames, and this excludes a unique definition of causal order in accordance with the special principle of relativity. So a Euclidean spacetime model has to be excluded.

(b) of course works fine. In Galilei-Newton spacetime time is absolute and thus simply defining it as a oriented 1D manifold fixes the causal direction of time once and for all inertial reference frames

(c) of coarse also works with the usual caveat that there's a "limiting speed", and only "time-like" or "light-like" connected events can be in a cause-effect relation, which then is independent on the choice of the inertial reference frame. To the best of our empirical knowledge the "limiting speed" is the phase velocity of (plane) electromagnetic waves (with an upper limit of the "photon mass" being $10^{-18} \, \text{eV}$.

For more on this very illuminating derivation of the possible spacetime models having global inertial reference frames, see

V. Berzi and V. Gorini, Reciprocity Principle and the Lorentz Transformations, Jour. Math. Phys. 10, 1518 (1969),
https://doi.org/10.1063/1.1665000

 

[PeterDonis]

Wouldn't imaginary time be a fifth dimension?

No. It's just multiplying time by $i$ to make the Minkowski interval look like the Euclidean distance in a 4D space. It's a mathematical trick which, as the MTW quote @Hill gave shows, is only workable in flat spacetime and is of limited value even then.

 

[vanhees71]

As I tried to say already above, "imaginary time" has some mathematical applications. The one I mentioned above is the Matsubara formalism of equilibrium quantum many-body theory. The other is the use of Wick rotation in vacuum relativistic QFT, where you can prove some things related to, e.g., renormalization easier than working in Minkowski space. The problem then, however, is to analytically continue in the right way back to real time. In any case it's a mathematical/calculational tool rather than something with direct physical meaning.