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If the fundamental bits of energy in the...
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You are currently reading a thread in /sci/ - Science & Math

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If the fundamental bits of energy in the universe (those "vibrating strings") are non-divisible and thus homogeneous how it is that they are understood to vibrate differently from one another? How is such a difference brought about?
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>>7779112
bump
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>>7779112
The picture of a vibrating string is better suited as just an analogy when dealing with the quantum mechanical behavior.

I will try to show what is going on with a toy model, bosonic string theory.

So what we mean by the vibrating is the different levels of energy excitations of the string states.

So if we take a look at closed string world sheet model, we describe the string by functions which embedded the worldsheet into the background spacetime.

[math] {X_R}^\mu \left( {\sigma ,\tau } \right) = \frac{1}{2}{x^\mu } + \frac{1}{2}{\ell _s}^2{p^\mu }\left( {\tau - \sigma } \right) + \frac{i}{2}{\ell _s}\sum\limits_{n \ne 0} {\frac{1}{n}\alpha _n^\mu {e^{ - 2in\left( {\tau - \sigma } \right)}}} [/math]

(and similarly for left movers)

The important part we want to use in these equations, are the fourier modes [math] \alpha _n^\mu [/math] .

We want to canonically quantize the string worldsheet model by promoting these fourier modes to linear operators.

So we define, in terms of these modes, creation&annihilation operators.

[math] a_n^\mu = \frac{1}{{\sqrt n }}\alpha _n^\mu [/math]
[math] a{_n^\mu ^\dagger } = \frac{1}{{\sqrt n }}\alpha _{ - n}^\mu [/math]

We can then derive the canonical commutation relations: [math] \left[ {a_n^\mu ,a{{_m^\nu }^\dagger }} \right] ={\eta ^{\mu \nu }}{\delta _{mn}} [/math]


We now have a quantized description of our string (for a flat spacetime and conformally flat worldsheet).

Ok so what we mean by the vibrational levels are the different energy levels obtained by acting on the string ground state with these operators.

i.e. define a the ground state as [math] \left| 0 \right\rangle [/math] such that [math] a_n^\mu \left| 0 \right\rangle = 0 [/math]

Then the different vibrational levels are the generic states obtained by acting on this ground state with the creation operators...
[math] \left| S \right\rangle = a{_{{n_1}}^{{\mu _1}}^\dag }...a{_{{n_m}}^{{\mu _m}}^\dag }\left| 0 \right\rangle [/math]
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>>7779657
** Fixed code lines:


[math] a_n^{\mu \dagger } = \frac{1}{{\sqrt n }}\alpha _{ - n}^\mu [/math]

and


[math] \left| S \right\rangle = a_{{n_1}}^{{\mu _1}\dagger }...a_{{n_m}}^{{\mu _m}\dagger}\left| 0 \right\rangle [/math]
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>>7779657
>>7779662
Yeah /sci/ will totally understand this.
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>>7779657
>>7779662

Not OP, but thanks for this post anon. I don't have the background to fully appreciate this (I've only taken one semester of quantum mechanics), but do you mind explaining your first equation and what precedes it a bit more?
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>>7779831
The action of this very simple string theory takes the form:

[math] S = \frac{T}{2}\int {{d^2}\sigma } {\eta _{\mu \nu }}{\eta ^{\alpha \beta }}{\partial _\alpha }{X^\mu }{\partial _\beta }{X^\nu } [/math]

The classical equation of motion which is derived from this action is a wave equation:

[math] \left( {\frac{{{\partial ^2}}}{{\partial {\sigma ^2}}} - \frac{{{\partial ^2}}}{{\partial {\tau ^2}}}} \right){X^\mu } = 0 [/math]

So that first equation in my original post is a solution of the wave equation in the form of a fourier series.

The [math] {x^\mu } [/math] are center of mass position coordinates.

The [math] {\ell _s} [/math] is the string length scale.

The [math] {p^\mu } [/math] is the total string momentum.

And the sigma and tau are the spacial and temporal coordinates of the 2D string worldsheet.
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>>7779846
>very simple
>string theory
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