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
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.
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