Complex vector X, X=point in 3D phase space, X*X = 0.

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  • #1
Spinnor
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Main Question or Discussion Point

Let X=(x1, x2, x3) be an element of the vector space C^3. The dot product of X with itself, X·X, is (x1x1+x2x2+x3x3). Note that if x1=a+ib then x1x1=x1^2 = a^2 - b^2 + i(2ab), rather that a^2+b^2, which is x1 times the conjugate of x1.

Let the real part of C represent the position of a particle in R^3 under the influence of 3D harmonic oscillator potential. Let the imaginary part of C represent the velocity of this particle.

What possible paths of a point particle in R^3 subject to a 3D harmonic oscillator potential, if any, satisfy X·X = 0 ?

Possible paths of a point particle in R^3 subject to a 3D harmonic oscillator potential lie in a plane?

Thank you for any suggestions on how you would solve this.
 

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  • #2
HallsofIvy
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Let X=(x1, x2, x3) be an element of the vector space C^3. The dot product of X with itself, X·X, is (x1x1+x2x2+x3x3). Note that if x1=a+ib then x1x1=x1^2 = a^2 - b^2 + i(2ab), rather that a^2+b^2, which is x1 times the conjugate of x1.
So you are using a "non-standard" dot product that is NOT an inner product over the vector space?

Let the real part of C represent the position of a particle in R^3 under the influence of 3D harmonic oscillator potential. Let the imaginary part of C represent the velocity of this particle.

What possible paths of a point particle in R^3 subject to a 3D harmonic oscillator potential, if any, satisfy X·X = 0 ?

Possible paths of a point particle in R^3 subject to a 3D harmonic oscillator potential lie in a plane?

Thank you for any suggestions on how you would solve this.
 
  • #3
Spinnor
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So you are using a "non-standard" dot product that is NOT an inner product over the vector space?

I think so. With such a definition, in The Theory of Spinors by Cartan, spinors "pop" out.
 

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