The Unruh Effect

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The Unruh Effect is very interesting and not very well known outside of the field of physics. Simply put, and accelerated observer will notice a thermal “bath” of particles arising from the vacuum. The “Unruh Temperature” of these particles will be

{\displaystyle T={\frac {\hbar a}{2\pi ck_{\mathrm {B} }}},}

where c is the speed of light, k is Boltzmann’s constant, h is planck’s (reduced) constant and a is the acceleration of the observer.

There is a famous sort of cartoon picture that explains where this effect comes from. According to this “cartoon,” particle-antiparticle pairs are constantly being produced in the vacuum, appearing for a brief time and annihilating each other soon after. However, because our observer is accelerating, they see an event horizon beyond which they can see no events, exactly analogous to a black hole. In the case of the black hole, this event horizon arises from a curvature of spacetime so intense that not even light can escape it. In this case, our observer is accelerating so light from too far away will never catch up, as at a certain point the observer will be approaching near-light speeds. Because of this horizon, if a particle-antiparticle pair is created right on the the horizon, the observer will only see one of them, intercepting the particle before it has time to be annihilated.

This cartoon picture, however, has nothing to do with the actual calculation of the effect. In fact, it’s not clear at all, looking  at the calculation, what it has to do at all with this picture.

Here is how the calculation actually plays out.

First, one must transform into the coordinates of the moving observer. In a particularly nice choice of coordinates, this is a conformal transformation, making calculations nice. One ten can calculate how the observer’s annihilation and creation operators of the field are transformed from the vacuum ground state. Finding the Bogloliubov coefficients of this transformation, one finds that the spectrum of particles produced is exactly equal to black body radiation with an analogous temperature as expressed above.

But this calculation makes it very unclear why the final spectrum should look so much like black body radiation. It just sort of pops out, and you are free to interpret it as some sort of an “effective temperature,” if you like. However, it really has nothing to do with thermodynamics, as we did not even define entropy, or anything like that. It is simply not a calculation from statistical mechanics.

Where, then, is the origin of this effect? Why must it take the form it does? Well, another explanation, doing this calculation with path integrals, does a sort of transformation of the Lagrangian being integrated over and makes it take the same algebraic form as the partition function from statistical mechanics. So that at least motivates it.

I think it’s very interesting how subtle this discussion of Unruh radiation is. It’s rarely explained well, and often not made clear what interpretations are helpful and which are a little misleading.

 

Introduction to Quantum Effects in Gravity

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“Introduction to Quantum Effects in Gravity” by Mukhanov and Winitzki is a book that I spent a significant amount of time reading pretty carefully. I wanted to say a few words about this book. It introduces quantum field theory very carefully, quantizing a massive scalar field in empty space, then going on to do it in certain space times. It shows how to find the vacuum state and introduces Bogoliubov transformations of that state as excited states. From the coefficients of these transformations come the intensity of particles produced of different wavelengths. Detailed calculations are done for the Unruh Effect (which I will talk about in the next post) and Hawking Radiation.

What’s nice about this introduction is how self contained it is. Bogoliubov transformations are introduced early, well motivated, and used throughout the entire book to explain many different phenomena. The particle interpretation of quantum field theory is made pretty clear, with particle densities of certain frequencies related to Bogoliubov coefficients.

For this reason I recommend this book to anyone interested in introducing themselves to the subject.

Interestingly, the presentation of the book demonstrates how little background you need to understand the basics of pretty deep physics. I often hear people worry about specialization in the sciences. Perhaps physics is getting “too deep” and it takes so long for physicists to learn about one little subject, upon which a mountain of research has been done, that they become unable to effectively do their job. I think this fear is a little misplaced. While it is definitely true that the gestation time for a physicist is very long, the more research is done on a subject, the better it is understood within the context of all other physics, and the better taught it is. It takes students far less time to learn general relativity now than soon after it was developed because it is understood so thoroughly now that the pedagogy significantly aids the teaching. “Introduction to Quantum Effects in Gravity” is a good example of a pedagogical introduction to a pretty opaque subject.

 

What is a Quantum Field?

What exactly a “Quantum Field” is confused me for a long time. Let me attempt to clear up some misconceptions.

When you picture a quantum field, you might picture something like this:

self-reproducing-universe-one-scalar-field-4

Some field in over all of space that evolve in time. Perhaps at the points where the value of the field is higher, we say that there is a particle there.

This is completely false. What that picture would represent is regular field theory. In regular theory you have just that, a field throughout all of space that evolves in time.

But in quantum field theory what you are studying is much more subtle. It is not a field throughout all of space-time but rather a wave-function over all possible fields.

Basically, in regular quantum mechanics, a particle has different probabilities to be at different points in space. But in quantum field theory, fields have different probabilities to be in certain “configurations.” (The picture above is an example of one such configuration.)

This means that a field has no one actual configuration. As such, you can’t actually ever know what the value of the field is as there is always some quantum uncertainty. Each little part of the field is subject to it’s own analogous version of the Heisenberg uncertainty principle.

This has some complications. For one, the number “degrees of freedom” of a quantum field is infinite. Creating a mathematical framework to easily calculate properties about these quantum fields is the first step of learning quantum field theory.

One major complication is what exactly a “particle” as we experience them is. This is a very subtle issue, but saying that a particle is a “spike” in a possible field configuration is pretty accurate.

Quantum mechanics is what happens when you “quantize” a particle. Quantum Field Theory is what happens when you “quantize” a field. What is interesting is that in Quantum Mechanics, particles behave a lot like waves, but in Quantum Field Theory, waves behave a lot like particles. It is a strange twist.

Three ways to find a straight line

Three ways to find a straight line

The geodesic equation is as follows:

The details aren’t important, but this equation lets you find the straight lines on a curved manifold. I have seen three derivations of this equation and thought I would share them here. Each derivation sheds some new light on the equation.

1. Minimize Distance

If you have some path on your curved manifold from one point to another, you can measure the distance of your path. If you want this path to be a local minimum (meaning all nearby paths are longer) then you can derive the geodesic equation using the Euler Lagrange equation. This derivation shows you that “straight” lines are the ones that minimize distance in a very global sense.

2. Parallel Transport

If you have a curved manifold you usually picture it sitting in some higher dimensional space. A warped piece of paper sits nicely in regular three dimensional space. A warped higher dimensional manifold might also sit nicely in an even higher dimensional space. If you have, say, a little ant moving in some direction on this manifold, embedded in a higher dimensional space, it will take a step forward along it’s velocity vector. Now, note that it’s velocity vector is tangent to the manifold. If the ant moves straight along the velocity vector it will fall off the manifold. Instead, the ant must be constrained to move on the manifold. Therefore it’s velocity vector must change a little bit when it moves. The amount it’s velocity vector changes is given exactly by that upside down L figure in the equation. Through the process of taking a step forward as best the ant can along its velocity vector, “parallel transporting” its velocity vector along this path, then taking another step forward, it then moves again along this new transported velocity vector.

This is a beautiful proof because it gives this dynamical local description of what’s going on. A trajectory is generated by a single initial velocity vector tangent to the manifold and keeps following this vector, with the vector constrained to be tangent to the current point.

3. The Equivalence Principle

This is the physics proof of the above equation. It imagines the trajectory as the path of a falling observer in a space time manifold.  Einstein’s equivalence principle states that for any free falling observer there should be a reference frame in which the observer is in regular flat (Minkowski) space time. In this reference frame, the observer moves at a constant velocity along a straight line in space time. Then, doing a coordinate transform from that inertial reference frame into your preferred coordinates, out pops the geodesic equation.

This derivation is beautiful for many reasons, of which I will list a few. For one, it gives beautiful physical intuition. There is some reference frame in which we know thing move along straight lines and the equations are easy. It stands to reason that when we transform coordinates, these equations are still “straight.” Furthermore, it shows how the motion of particles is guided completely by the curves of space time. The particle is really just moving in a constant velocity in a straight line in its inertial frame!

Conclusion
That these three derivations all fit together show how beautiful physics is. All these ideas are completely different ways of getting at the exact same thing.