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I’m currently reading Max Tegmark’s book Our Mathematical Universe. In talking about the number of space and time dimensions we live in, he briefly mentions tachyons existing in one space dimension and three time dimensions.

I guess my questions are as follows:

What exactly are tachyons and how can they have imaginary mass?

How widely accepted are they in the science world?

What would reality with 3 time dimensions and one space dimension even be? Is it something the human mind can even imagine?

In: Physics

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Tachy means fast, in this case, faster than light.

All massless particles move at the speed of light, and particles of negligible mass move at significant fractions of the speed of light. This is called c. To speed up a particle to the speed of light requires infinite energy. It’s like an exponential decay function flipped upside down, for a given mass, you can never reach the speed of light but can get ever so close.

Negative mass is impossible, but you can think of mass as a function of some other mystery quantity, i.e. the square root of it (M^0.5). If that quantity is positive, the mass is positive, and therefore cannot reach the speed of light. If it’s zero, then it is at the speed of light (sqrt of 0 is 0), and if it’s negative, then the mass is imaginary, and moving faster than the speed of light. Tachyons are particles moving faster than light due to this property.

This isn’t necessarily an accurate explanation, but one you can get behind using some early high school math.

As others said, they are faster than light particles. The relativistic equations set a speed limit in the universe equal to the speed of light, but it turns out you can solve them at speeds greater than the speed of light. The catch is it would take infinite energy for a tachyon to slow down to the speed of light and infinite energy for a normal particle to speed up to the speed of light.

All of this means that likely it’s an error in the relativistic equations and tachyons can’t exist. The standard model has tachyon condensation which prevents them from existing.

If you look at the equations for energy of a moving particle in special relativity, you find that there is no solution for a particle with mass moving at the speed of light. The mass of the particle (and so the energy) goes to infinity as you approach it. However you can solve the equations for particles moving faster than the speed of light. Those hypothetical particles are called tachyons. They have never been observed, nor has there been any experiment that would require one to explain an observation. But if you plug a number for velocity > c into the relativistic kinetic energy equation you have to take the square root of a negative number, so the kinetic energy is a multiple of i which means is is imaginary. The only way for that to be possible is if the rest mass is imaginary.

As for the dimensions, they don’t exactly have 3 time dimensions and 1 space dimension. By solving other relativistic equations for them you find 3 of their dimensions have properties similar to the dimension we call time and 1 dimension has properties similar to the ones we call space. I don’t recall what those properties are, it was a long time ago that I studied this. But I can’t imagine what that would be like.

I’m not exactly sure about the whole one space dimension, 3 time dimensions, but the idea of tachyons (specifically tachyonic fields) can actually be somewhat easily understood. So imagine a sombrero, and say that I put a marble on the sombrero. This marble is going to be a lot happier (more stable) sitting in the brim of the sombrero than on the very tip of it. In fact, if the marble is sitting on the very tip, it will only take a very slight nudge to make it fall down to the brim.

So now, we can draw analogies between the marble/sombrero and the energies of vacuua of fields. In this case, the height of the sombrero gives the energy, and where I place the marble defines the actual vacuum state. So, in the brim of the sombrero, the vacuum is in its lowest energy state and is stable. However, at the tip of the sombrero, the vacuum is unstable to decay. These unstable vacuum configurations are exactly what we would call “tachyonic” field configurations.

So how do these have negative mass? Okay, well unfortunately the only way to see this is really through the math. A good function which describes this sombrero shape is something like a*x^4 – b^2 *x^2 where x here is the configuration of the vacuum. Now, if we care about the mass of particles which appear from this field, we have to look at small fluctuations away from the vacuum. The peak of the “sombrero” is at x=0, so if we consider fluctuations dx away from this vacuum, given by x+dx, the energy of the fluction is just a*dx^4 – b^2 *dx^2. We typically associate the mass of a particle with the square root of the coefficient of the dx^2 term, but here, we can see that the coefficient is negative! The only way for that to happen is if the “mass” is purely imaginary.

Now, if we fluctuate about the lowest point of the “sombrero” instead, at x=sqrt(2b^2 /(3a)), we find that the coefficient of the dx^2 term is 3b^2 which is positive! So if the vacuum sits in the unstable state, it produces tachyonic particles with imaginary “mass” whereas if the vacuum is happy in its stable state, it produces particles with real mass. This is actually an incredibly important thing that happens in many theories, including that of the Higgs boson.

The word tachyon just refers to any theoretical particle that travels faster than light. In the same way that any particle with mass can only ever travel below the speed of light, a tachyon would only ever be able to travel *faster* than light.

Tachyons, as far as we know, do not exist, nor are they theorized to exist. Certain mathematical models don’t rule them out, but there are other reasons that we generally believe tachyons are impossible, and it’s generally accepted that tachyons do not and cannot exist, and they’re really just a mathematical quirk.

As for what a universe with 3 dimensions of time and one dimension of space would be like, no we can’t possibly imagine or speculate what that universe might be like or even if it would be stable enough to exist.