Specific Question about Susskind's Lecture on Tachyons

In summary, Susskind discusses the case of massless photons in 4 dimensions and the implication of having only 2 polarization states for preserving Lorentz invariance. He uses the example of a bosonic open string in D space-time dimensions to demonstrate that a massless photon would have (D-2) polarizations. The notation used by Susskind corresponds to the creation and annihilation operators for different excitations of energy. He concludes that the energy and squared mass of the first excited state must be zero, leading to the conclusion that the ground state must have a negative mass squared.
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I may have poorly titled this post, since the lecture I'm talking about isn't just about tachyons, really. What I'm referring to is what Leonard Susskind says between the 50-60 minute mark:

http://www.youtube.com/watch?v=gCyImLu0HSI&feature=youtu.be&t=46m30s

In the video, Susskind concludes that there are the only two polarization states of the first excited state of the open string, so then this string must be massless, like a photon. For this to happen, the ground state must have a negative mass squared.

Why must [tex]m_0^2 +1 =0[/tex] and how do we know that the a's and the b's represent the only two polarization states? I understand that if there are only two polarization states, then the particle must be massless to preserve Lorentz invariance, but the rest is confusing to me.

Can someone clear this up for me? I have no formal exposure to QM (and I apologize for this); regardless, I will try to decipher any technical answer given. Thanks so much.
 
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The example that Susskind takes is the simplified case of massless photons in [itex]4[/itex] dimensions. A massless photon in [itex]4[/itex] dimensions has only [itex]2[/itex] polarizations (so are [itex]a_i/a_i^+[/itex] and [itex]b_i/b_i^+[/itex], which are the anihilation/creation operators, for different excitation of energy [itex]i[/itex], corresponding to the 2 possible polarizations X and Y), but in a [itex]D[/itex] space-time dimension, a massless photon has [itex](D-2)[/itex] polarizations. For instance, in the bosonic open string, the coherent dimension is [itex]D=26[/itex], so you have [itex]24[/itex] possible polarizations. But the logic is the same, the excitations are vectors, that is from the ground state [itex]|0\rangle[/itex], and applying creation operators for the lowest energy excitation, we have states [itex](\alpha^\mu)^+_1 ~|0\rangle[/itex] with [itex]\mu = 1...D-2[/itex]. Susskind notations correspond to [itex]a = \alpha^1, b= \alpha^2[/itex]. The subscript [itex]_1[/itex], in [itex](a^\mu)^+_1[/itex] means that we consider only the lowest energy excitation. The energy of these excited states is [itex]m_0^2+1[/itex], and it is also the squared mass of this state, which must be zero, so we have [itex]m_0^2+1=m^2=0[/itex]
 
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What is the concept of tachyons in Susskind's lecture?

In Susskind's lecture, tachyons are described as hypothetical particles that travel faster than the speed of light. They are a theoretical concept in physics that has not yet been proven to exist.

How does Susskind explain the properties of tachyons?

Susskind explains that tachyons have imaginary mass and can only travel faster than the speed of light. They are also believed to have the ability to travel backwards in time, which goes against the principles of causality in physics.

What are the implications of tachyons?

The existence of tachyons would challenge many fundamental principles of physics, such as the theory of relativity and the concept of causality. It would also open up the possibility of time travel and communication beyond the speed of light.

Can tachyons be detected?

So far, there is no evidence or experimental data to support the existence of tachyons. They are purely a theoretical concept and cannot be detected using current technology.

How do tachyons relate to other theories in physics?

Tachyons are often discussed in relation to other theories, such as string theory and quantum mechanics. They are also connected to the concept of imaginary time and the multiverse theory.

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