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As an easy example consider the complex vector space $\Bbb C^2$. We can consider $\Bbb C^2$ as vector space over $\Bbb R$ and thus have the four basis vectors

$$ \hat e =\{(1,0), (i,0), (0,1), (0,i)\} $$

because $\Bbb C^2$ has four real degrees of freedom.

An arbitrary element $\phi$ of $\Bbb C^2$ can then be written as

$$ \phi = \sum_{i=1}^4 \phi_i \hat e_i \quad \text{ with } \quad \phi_i \in \Bbb R $$

The corresponding conjugate element $\bar \phi \in \bar{\Bbb C}^2$ is

$$ \bar \phi = \sum_{i=1}^4 \bar \phi_i \bar {\hat e_i} = \sum_{i=1}^4 \phi_i \bar {\hat e_i} \quad \text{ because } \quad \phi_i \in \Bbb R $$

For example,

$$ \phi = (a+ib,0) \text{ with } \quad a,b \in \Bbb R \rightarrow \bar \phi = (a-ib,0)$$

Do we really have new degrees of freedom in $\bar \phi$, because the element $ (a-ib,0)$ is an element of $\Bbb C^2$, too?

Thus my question: Do we have four or eight independent degrees of freedom in the complex vector space $\Bbb C^2$, together with its conjugate vector space $\bar{\Bbb C}^2$?

Community
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jak
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1 Answers1

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$\Bbb C^2$ is a $4$-dimensional space (i.e. a space with "4 degrees of freedom"). The mapping $\phi \mapsto \bar \phi$ is a (real-)linear transformation from $\Bbb C^2$ to $\Bbb C^2$ (that is, $\overline{\Bbb C^2} = \Bbb C^2$). The existence of this map does not change the dimension of the space.

Ben Grossmann
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  • Thanks for your answer. In what sense do you mean $\overline{\Bbb C^2} = \Bbb C^2$? This confused me a bit, because in physics the basis vectors of $\Bbb C^2$ are called particles and the basis vectors of $\overline{\Bbb C^2}$ are called anti-particles. If we consider $\Bbb C^2$ as vector space over $\Bbb C$, we have two basis vectors and thus two "complex particles". Thus is it superfluous to talk about "anti-particles" in $\overline{\Bbb C^2}$, because these degrees of freedom, like $\bar \phi$ in the original question are already contained in $\Bbb C^2$? – jak Aug 25 '15 at 14:42
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    I don't know about the physics, but from what you've said it seems that the "particles" and "anti particles" are just two different bases for the same space. – Ben Grossmann Aug 25 '15 at 14:53