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Canonical invariance vs. Lorentz invariance

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1. Homework Statement
I have an assignment to prove that specific intensity over frequency cubed is Lorentz invariant. One of the main tasks there is to prove the invariance of phase space [itex]d^3q \ d^3p[/itex] and I am trying to prove it with symplectic geometry. I am following Jorge V. Jose and Eugene J. Salentan: Classical dynamics.

2. Homework Equations
[tex]\omega = dq^{\alpha} \wedge dp_{\alpha}\\
v = d^3q \ d^3p = \frac{1}{n!} \omega^{\wedge n} = \frac{1}{n!} \omega \wedge \omega \wedge \cdots \wedge \omega = dq^1 \wedge dp_1 \wedge \cdots \wedge dq^n \wedge dp_n,[/tex]
of course in my case [itex]n=3[/itex].

3. The Attempt at a Solution
The book says that this [itex]v[/itex] is invariant under canonical transformations, because [itex]\omega[/itex] is. I am now wondering if this is enough also for the Lorentz invariance?
 

jambaugh

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It would be sufficient if Lorentz transforms manifest as canonical transformations in phase space. Do they? If not then you'll need to show it directly. To answer the question I think you cannot avoid explicitly dipping into the details of how the Lorentz group acts on phase space. So get too it.
 
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I tried now in another way. By the definition of forms and change of coordinates (the Jacobi matrix) which is for [itex]p[/itex] done also in Goodman. I write Lorentz matrix for boost in [itex]x[/itex] direction, which is


[tex]\Lambda_{\mu}^{\nu} =

\begin{pmatrix}

\gamma & \gamma \beta & 0 & 0\\

\gamma \beta & \gamma & 0 & 0\\

0 & 0 & 1 & 0\\

0 & 0 & 0 & 1

\end{pmatrix}, [/tex]

so that the transformation is

[tex]

x'_0 = \gamma(x_0 + \beta x_1),\\

x'_1 = \gamma(x_1 + \beta x_0),\\

x'_2 = x_2,\\

x'_3 = x_3.

[/tex]




I know that [itex]v = p/E = p^1/p^0[/itex]. Following Goodman, I can get the [itex]d^3p/p^0[/itex] invariance, but when I try to do Jacobain determinant for [itex]d^3x[/itex] in the same way, I get

[tex]\begin{vmatrix}

\frac{\partial x'_1}{\partial x_1} & \frac{\partial x'_1}{\partial x_2} & \frac{\partial x'_1}{\partial x_3} \\

\frac{\partial x'_2}{\partial x_1} & \frac{\partial x'_2}{\partial x_2} & \frac{\partial x'_2}{\partial x_3} \\

\frac{\partial x'_3}{\partial x_1} & \frac{\partial x'_3}{\partial x_2} & \frac{\partial x'_3}{\partial x_3}

\end{vmatrix} = \begin{vmatrix} \gamma (1 + \beta \frac{\partial x_0}{\partial x_1}) & \gamma \beta \frac{\partial x_0}{\partial x_2} &\gamma \beta \frac{\partial x_0}{\partial x_3}\\
0&1&0\\
0 & & 1
\end{vmatrix} = \gamma(1+\beta \frac{\partial x_0}{\partial x_1}). [/tex]

I then assume that [itex]\frac{\partial x_0}{\partial x_1} = 1/v[/itex], since [itex]x_0 := t[/itex] and also [itex]v = \frac{p_1}{p_0}[/itex], so the final determinant is equal to [itex]\gamma(1+\beta \frac{p_0}{p_1})[/itex], which can be rewritten as
[tex]\frac{p'_1}{p_1},[/tex]
which is clearly not what I want. Should be [itex]\frac{p_0}{p'_0}[/itex]. Where am I mistaken?
 

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