inverse temperature and imaginary time

[scruffy]

In addition to relativity, imaginary time is also important in quantum mechanics. It's the basis of the time dependent Schrodinger equation, which describes the "evolution" of a quantum system.

The time dependent equation is:

ih (dW/dt) = H W

where W is the wave function and H is the Hamiltonian that represents the total energy of the system (and h is Planck's constant).

This usage in quantum mechanics is çonsiderably more sophisticated than its cousin in relativity. In relativity, we're simply changing a metric (it's almost like a coordinate transformation).

However here we arrive at the answer using probability amplitudes which are used to calculate the contributions of all possible paths to an event at a point in time (usually the event is a measurement, that's why the point in time is fixed).

4.4: The Time-Dependent Schrödinger Equation

While the time-dependent Schrödinger equation predicts that wavefunctions can form standing waves (i.e., stationary states), that if classified and understood, becomes easier to solve the time-…

chem.libretexts.org

 

[scruffy]

You can apply this trick of Wick rotation to any variable, it doesn't have to be time. You can treat Planck's constant like an imaginary temperature. One clever application is to use a Wick rotation on the Lagrange multipliers of a constrained optimization problem. The principle of least action is one such problem.

But you have to be careful. It doesn't really make sense to maximize a complex function. So you have to look for stationary points where the derivative is 0. And if you have a logarithm term like if you're working with entropy, log(x) now becomes log(ix) which is periodic, adding 2πi to any answer will get you an equally valid answer.

For a measurement or any other quantum interaction, you can reciprocate a time series of actions against a sequence of histories (essentially these are complementary reference frames). This gives you "all possible paths" and the associated energies and times. You can use the actions to get the probabilities.

All this quantum stuff is relativistic, it's happening at the speed of light. The equations get real hairy in Minkowski space, so you Wick-rotate into complex space, solve, and Wick-rotate back.

 

[ReinyDays]

WTF are you going on about?

We're not talking about shear rotations.

Address the issue or go away.

I'm not going away ... go to another message board if you don't want to deal with my questions ... you can spread your falsehoods someplace else ... you programmed your AI wrong and now it's spewing nonsense ...

What is a shear rotation and why is it important to the OP? ... you don't know, so everything you say about Wise rotations is wrong ... especially treating complex numbers as vectors ... that's not what Wise is doing ...

Every Wise rotation is shear locally ... so we are talking about shear rotations that are within 100 million light-years ...

 

[scruffy]

I'm not going away ... go to another message board if you don't want to deal with my questions ... you can spread your falsehoods someplace else ... you programmed your AI wrong and now it's spewing nonsense ...

You're just trolling. You're trying to figure out what we're talking about.

What is a shear rotation and why is it important to the OP? ... you don't know, so everything you say about Wise rotations is wrong ...

What is a Wise rotation?

especially treating complex numbers as vectors ... that's not what Wise is doing ...

We haven't started talking about vectors yet.

Every Wise rotation is shear locally ... so we are talking about shear rotations that are within 100 million light-years ...

Where are you getting "shear rotations" from? There's no shearing going on here. The demons are poking nonsense into your brain.

We are specifically preserving angles. That's what "conformal" means. No shearing.

 

[scruffy]

Here is an excellent introduction to the mathematical treatment of "all possible paths".

Path integral formulation - Wikipedia

en.wikipedia.org

How do you determine whether a path is possible? Before Feynman, before Dirac, there was Norbert Wiener who studied Brownian motion. Wiener gave us a formula for the expectation of any path. All subsequent work is based on his model.

The Wiener equation assumes the velocity of a particle fluctuates randomly, so it becomes imprecise at very short time scales (dt). A Langevin equation however is based on accelerations (second moments).

Langevin equation - Wikipedia

en.wikipedia.org

Brownian motion tries to answer the question "where will the particle go next", and this is the same question being asked by Schrodinger and Feynman. Only, in a quantum field, you have to answer that question for each point.

What do all these methods have in common?

They all use infinitesimal Wick rotations to propagate the system forward. In particle physics a possible path is called a "propagator".

This usage of the Wick rotation is different from its macroscopic usage.

 

[scruffy]

Oh - I should point out that the Lagrangian is Lorentz-invariant whereas the Hamiltonian is not.

 

[scruffy]

So let's get into the interesting part.

We talked about moments, and the moment generating function. In probability theory, there is another important entity called the characteristic function.

The characteristic function is the Fourier transform of the probability distribution. So while the moment generating function is

E[e ^ xt]

the characteristic function is

E[e ^ ixt]

The characteristic function is a Wick rotation of the moment generating function.

And you will remember Euler's formula which is

e ^ it = cos t + i sin t

The characteristic function is thus

cos xt + i sin xt

which is another way of saying that just like any other function, you can build any probability distribution from periodic functions of various frequencies, phases, and amplitudes.

 

[ReinyDays]

You're just trolling. You're trying to figure out what we're talking about.



What is a Wise rotation?



We haven't started talking about vectors yet.



Where are you getting "shear rotations" from? There's no shearing going on here. The demons are poking nonsense into your brain.

We are specifically preserving angles. That's what "conformal" means. No shearing.

What are you doing then? ... this is a message board, not your personal blog ... I'm asking questions, and you're refusing to answer ... maybe you're not a troll but certainly you're a bastard ...

You know perfectly well I'm talking about rotations as used in Analytical Geometry ... and a shear rotation is a rotation in a Euclidian space ... you're pretending to not know this, which is the true troll ... or maybe you haven't taken any college math classes ... and you don't know we're rotating tensors with complex magnitudes ...

But the AI you're using will still tell you what you want to hear ... that's how it's programmed ... must be comforting to you ...

Do you go to college at all? ... didn't you take a class called "English Composition"? ... let me remind you, you write to your audience ... and here, we haven't taken differential geometry ... so you need to make this a lot simpler ...

You never answer this ... did you go to college? ...

 

[scruffy]

I'm talking about rotations as used in Analytical Geometry ...

The complex domain is a weird place.

It's not like the Euclidean geometry you learn in high school.

In complex space, a rotation is a change in phase. It's like taking a sine wave and sliding it over to the right or left.

A phase change in quantum mechanics corresponds with a change in the probability amplitude. You're not rotating the electron, that's something completely different.

 

[scruffy]

Here, I'll give you a freebie.

The key piece of math in a Wick rotation is analytic continuation.

In this context, it means you're adding a variable, essentially "lifting" your equation into a higher dimension.

Let's say we have a function, z = f(x). We can plot it on a graph of z vs x, yes?

Now we're going to add a "dummy" variable, call it y. So our function will become z = f(x,y). But to start out, we'll always leave y to be 0, so it has no influence on anything.

Now we can no longer show our function on a graph in two dimensions, right? We need three. A graph like this:

If y is always 0, we can take a "slice" through this new graph along the plane y=0 and we recover the original two dimensional graph.

But if we allow y to vary, maybe we get something more interesting, like the parabola above - depending on our function f. The height of the image on the z axis now responds to both x and y.

It's the exact same thing with continuation in the complex domain. Except now instead of y we have i.

The complex domain is a convenient way to handle periodic functions. Like sine and cosine. Which are all the ones involved in physics. For convenience in representation we use 2π as the periodicity.

From Euclid, we start like this:

Take a real interval, call it 0 to 2π. It's a line segment, a piece of string. Now compactify your string by joining the ends - it becomes a circle. If you put a ball on the string it goes round and round instead of up and down.

By bending the string you are essentially "lifting" your straight line into an additional dimension, because your circle is obviously two dimensional, it has a radius and an angle, and you can translate those back into Cartesian x and y coordinates using an inverse polar coordinate transformation.

If we imagine putting a sine wave onto the string, instead of a ball, we can set it up so the period equals the string length, and that way as we travel along the sine wave on the x coordinate we end up rotating our angle on the circle while the radius stays constant.

A rotation of 90 degrees is equivalent to π/2, because it takes us 2π to go around the entire circle. But if we rotate (counterclockwise) by 90 degrees, then everything that used to be on the real axis (namely, our original function) is now on the imaginary axis. And what we have equivalently done is shifted the phase of our sine wave by 90 degrees, or in a manner of speaking, "moved time forward" by 1/4 of a period.

But you'll note now, that since we left our original y at 0, we can just multiply by i which has the same effect as rotating by 90 degrees, that is to say "moving time forward" by 1/4 period. This is the principle of a Wick rotation.

In the quantum world everything is periodic. Energy is frequency. And frequency is period, and period is an angle. Outside of some weirdo standing solutions, all your basic solutions to the Schrodinger equation are sines and cosines of varying frequencies, phases, and amplitudes.

The benefit for experimentation is that changing phase has the same effect as moving time forward. You can change the phases of quantum states with lasers and etc. This is part of how we got to "time crystals" and other oddball time related phenomena that are currently being investigated.

 

[ReinyDays]

The complex domain is a weird place.

It's not like the Euclidean geometry you learn in high school.

In complex space, a rotation is a change in phase. It's like taking a sine wave and sliding it over to the right or left.

A phase change in quantum mechanics corresponds with a change in the probability amplitude. You're not rotating the electron, that's something completely different.

Don't you mean the magnitude? ... and in-of-itself is neither Euclidean or non-Euclidean ... just like the real numbers, they're used in both ... obviously, you haven't been to college ... and you don't know the difference between scalars, vectors and higher-ordered tensors ...

Here, child ... a textbook discussion of what the AI is feeding you ... you may not have the wits to understand, but others might ... the TL;DR version is that calculating voltage and current independently during a phase shift is quite labor intensive ... much easier is creating a value that includes both and then mapping into the complex numbers ... the link above gives all the dirty details ...

Inverse temperature ... imaginary time ... just ChatGPT hallucinations ... when you take second year calculus this will be more clear ...

This is the best video I could find explaining what we're rotating ... and why ... it uses real numbers but we could use complex numbers just as easily ... just makes thing more difficult to explain ... we're not throwing house cats into black holes here, just accelerating them to the speed of light is all ... not cruel ...

 

[scruffy]

Don't you mean the magnitude? ...

No.

and in-of-itself is neither Euclidean or non-Euclidean ... just like the real numbers, they're used in both ... obviously, you haven't been to college ... and you don't know the difference between scalars, vectors and higher-ordered tensors ...

You're over complicating. You don't need Riemannian manifolds to understand this concept.

Here, child ... a textbook discussion of what the AI is feeding you ... you may not have the wits to understand, but others might ... the TL;DR version is that calculating voltage and current independently during a phase shift is quite labor intensive ... much easier is creating a value that includes both and then mapping into the complex numbers ... the link above gives all the dirty details ...

Looks like you're the one going to the internet for answers.

Inverse temperature ... imaginary time ... just ChatGPT hallucinations ... when you take second year calculus this will be more clear ...

Temperature is not an observable, it's a statistic.

This is the best video I could find explaining what we're rotating ... and why ... it uses real numbers but we could use complex numbers just as easily ... just makes thing more difficult to explain ... we're not throwing house cats into black holes here, just accelerating them to the speed of light is all ... not cruel ...

Solving physical configurations is hard enough. Solving them at the speed of light is harder. Solving them in a curved universe is still harder.

How will you account for the residual energies stored in configurations at 0 temperature?

 

[scruffy]

Potential energy is energy stored in a configuration. It still exists even at 0 temperature.

A configuration means potential energy can only be specified "relative" to something.

To get a complete description of potential energy you'd have to know all the other configurations, and then to understand what it means you'd have to know all possible configurations.

So you're left with the same problem, all possible paths or all possible states, we can figure out one from the other.

Fortunately, the concept of "all possible" has some common and well defined math. It turns out, you can do the same thing with an action that you can do with entropy and time, you can multiply it by i - and that has physical meaning because all the underlying functionals are periodic.

 

[scruffy]

In classical physics we want to minimize energy. Energy is something an entity "possesses", in terms of the number of quanta it's asked to carry. Natively the entity will seek its lowest energy state.

In mechanics there is action, which is energy * time (the number of quanta times the time the entity "possesses" them). The Planck constant has units of action. The classical Hamiltonian uses the "principle of least action", or more specifically stationary action, in the form of a derivative that's set to 0.

In quantum land however the action is a functional with a complex argument that has a real result in the form of a probability. The sum of all the probabilities has to add up to 1, so we have to normalize by "all" possible paths, the same way we use the partition function to characterize the entropy of an ideal gas. The form of the classical partition function Z is formally identical to "all" of Feynman's possible paths if we simply Wick rotate by ih. This makes the Schrodinger equation nothing more than a Wick rotated heat equation (a diffusion equation with a complex diffusion constant).

The idea that this transform should inherently involve probabilities is highly intriguing. Both selective and non selective versions of the outcome utilize a path between two endpoints, and sometimes the particular path matters and sometimes it doesn't. In quantum land we can even invert the reference frame and look at all the possible historical paths that might have led to the present state.

Wick rotations are powerful tools.

 

[scruffy]

So far we have the physical picture.

But the real prize is in biology. In neuroscience, and perhaps in AI.

Think about "decision making under uncertainty". Like, you're in a math exam, asked to solve a problem you've never seen before, and you have a finite amount of time to do it.

There's a thousand variations on this theme. You can train a monkey to look at a screen of moving dots, and perform an action whenever a significant (threshold) fraction of the dots move together, either in the same direction or at the same time, etc.

The logic involved in solving a problem involves endpoints and a series of actions. The actions are constrained by configurations, just like in physics. The biologists started studying this for real in the early 80's, when there was enough knowledge about brain function to start asking intelligent questions. And it turns out there are brain areas that specialize in finding and calculating "all possible paths" - all the ways you can get between two endpoints.

In a way, the deadline in a math exam is like a measurement in physics. It occurs at a fixed time. You either know your endpoints in advance, or a set of conditions triggers the second endpoint, and once it's known both endpoints are once again fixed.

When determining path, the logical decision making process is "piece-wise" stationary. There are usually intermediate goals that require intermediate steps - think of them like milestones - and we make choices using criteria like "easiest" or "fastest" or "most reliable". These are optimization problems just like minimizing energy. It all depends on how you treat the Lagrangian and Hamiltonian. The physical utility of this approach is apparent in the existence of the time "independent" Schrodinger equation.

So, this idea of analytic continuation ultimately results in the equivalent of information geometry. We have a library of probability distributions, and we're going to put them together in different ways (linear combinations of complex probability amplitudes) to map the likelihood of what happens when time moves forward.

 

[scruffy]

Another link with the brain:

There is phase coding of information, usually relative to a rhythm like theta or alpha. The phase coding usually occurs within a window of 100 msec or so, one period of the rhythm. This happens for example in the hippocampus with the theta rhythm. There are "place cells" that encode where the organism is relative to the receptive field center, with a burst of phase dependent activity. There are also "time cells" that encode the intervals between events the same way.

Any time we have phase relative to a periodic signal, we're automatically in the complex domain. So our phase encoded variable is like an angle and its real and imaginary parts can be recovered with cos and sin. We can apply a rotation operator to our angle to move time backward or forward, that is to say, change the phases of events with respect to the zero crossings of the periodic signals.

And, like in physics, the rotation isn't limited to just time, we can rotate around anything that's phase encoded. In the case of a place cell maybe that's distance from a landmark. So the "tape" of whatever happens in episodic memory as the organism approaches and then passes the landmark, is keyed to the phase of firing of the place cell.

We can easily reverse the direction of travel by rotating in the opposite direction. Essentially "play the tape backwards". This capability emerges automatically during compactification. If you take a linear segment of time and compactify it into a circle, you're now going round and round the circle. If you take this circular activity and project it back down on the real axis, you'll find that time is now moving in both directions. The top half plane takes you in one direction and the bottom half plane takes you in the other.

This capability is useful for the credit assignment problem in reinforcement learning. Which of the many preceding actions ultimately led to the reward? (Or punishment?) To answer you have to play the tape backwards.

 

[scruffy]

 

[scruffy]

Wick rotations in general curved spacetime:

https://arxiv.org/pdf/1702.05572

The video answers one of the complaints: a global symmetry turns into a local symmetry in the naive case, thus time can be rotated in both directions and casual structure disappears.

Well, it's not that it "disappears", it just goes to a different place. If you restrict the conformal map to the upper half of the complex plane the casual structure is preserved.

 

[scruffy]

Eureka!

The recalculation I refered to in a previous post is formally called the Osterwalder-Schrader theorem.

Not only does it calculate, it defines the precise conditions under which the Wick rotation will actually work.

Osterwalder-Schrader theorem in nLab

You'll probably want to take a look at this too:

Wightman axioms in nLab

The whole concept is based on the spin statistics theorem, which says bosons are social and fermions are not. Or put another way, no two spinors can occupy the same quantum state, which is the basis of the Pauli exclusion principle.

Spinors are the square roots of vectors. When you multiply a spinor by itself, you get a vector. To satisfy the physical constraints the spinors have to be Hermitian, that way the symmetries are preserved.

 

[scruffy]

And finally, there is a suggestive confluence of limits in algebra, physics, and biology.

In algebra, you have only four usable division algebras over the reals, including complex, quaternion, and octonion. In physics there are four dimensional relationships that don't exist in any other number of dimensions, and in biology you have four degrees of freedom for a nerve impulse. Four seems to be a magic number.

In all these cases you can separate variables, sometimes you can do 2+2 or 3+1. Complex numbers are commutative and associative and you get one essential symmetry which is the complex conjugate. Quaternions are associative but not commutative and you get full SO3 symmetry, and octonions are neither commutative nor associative but you get G2.