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$\newcommand{\k}[1]{\left | #1 \right\rangle }$ In his "The Principles of Quantum Mechanics", Paul Dirac states:

$$c_1\k A + c_2 \k B = \k R$$

Given two states corresponding to the ket vectors $\k A$ and $\k B $, the general state formed by superposing them corresponds to a ket vector $\k R$ which is determined by two complex numbers, namely the coefficients $c_1$ and $c_2$ of equation (1). If these two coefficients are multiplied by the same factor (itself a complex number), the ket vector $\k R$ will get multiplied by this factor and the corresponding state will be unaltered. Thus only the ratio of the two coefficients is effective in determining the state $R$.

Hence this state is determined by one complex number, or by two real parameters. Thus from two given states, a twofold infinity of states may be obtained by superposition.

This result is confirmed by the examples discussed in $§§$ $2$ and $3$. In the example of $§$ $2$ there are just two independent states of polarization for a photon, which may be taken to be the states of plane polarization parallel and perpendicular to some fixed direction, and from the superposition of these two a twofold infinity of states of polarization can be obtained, namely all the states of elliptic polarization, the general one of which requires two parameters to describe it. Again, in the example of $§$ $3$, from the superposition of two given translational states for a photon a twofold infinity of translational states may be obtained, the general one of which is described by two parameters, which may be taken to be the ratio of the amplitudes of the two wave functions that are added together and their phase relationship.

This confirmation shows the need for allowing complex coefficients in equation ($1$). If these coefficients were restricted to be real, then, since only their ratio is of importance for determining the direction of the resultant ket vector $\k R$ when $\k A$ and $\k B$ are given, there would be only a simple infinity of states obtainable from the superposition.


The above two paragraphs have been directly quoted from the aforementioned book.

The QUESTION is as mentioned in the title:

  1. What is the basic difference between twofold infinity and simple infinity? I mean, in mathematics, there is only one infinity as much as I knew.
  2. How is this difference interpreted in quantum mechanics? That is, does these things have a special abstract meaning here?
  3. And how does it even relate to states of polarisation? Can you suggest a suitable link to study these suitable states of polarisation like elliptical polarisation, circular polarisation,.. etc? I don't know what these are.

P.S. I found an answer to this question on functionspace but I couldn't clear my doubts using it. It was high above my level. So please try to use basic concepts of QM and, simple language and terms which I, a beginner, can understand.

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That is highly outdated language. What Dirac calls a 'twofold infinity of states' we would today call a two dimensional manifold. The distinct states $|R\rangle$ can be parametrized by two real parameters and form a two dimensional Bloch sphere.

If we restricted the coefficients $c_1, c_2$ to be real the condition that $|R\rangle$ must have unit norm would require the allowed states to form a one-dimensional manifold parametrized by one real parameter. For instance the parameter is $\theta$ and $$c_1 = \cos \theta,\quad c_2=\sin\theta$$

The connection with polarization is if we take $|A\rangle,|B\rangle$ to be two perpendicular linear polarized waves. Complex combinations of $|A\rangle,|B\rangle$ lead to elliptically polarized waves in general (see Jones vector), and Dirac's audience would know that these require two real parameters to describe.

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