Number of Linear Transformations: Probability, Quantum Mechanics and Beyond

Let $T= [t_{ij}]_{N\times N}$ . Since it preserves the fourth norm of the unit vectors, we must have $\sum_{i}t_{ij}^4 =1, \hspace{10pt} \forall j. \hspace{10 pt} (1)$ Now we will show that for any $i$ and $j \neq k$ , we have $t_{ij}^2 t_{ik}^2=0$ . For this, consider the vector $\vec{x}$ whose all elements excepting $j$ th and $k$ th elements are zero and $x_j=x_k= \frac{1}{2^{1/4}}$ , so that $||\vec{x}||_4=1$ . Now consider the $i$ th element of $T\vec{x}$ . It is given by $(T\vec{x})_i= \frac{1}{2^{1/4}}(t_{ij}+ t_{ik})$ . By the norm-preserving property of $T$ , we require $\frac{1}{2}\sum_i (t_{ij}+t_{ik})^4=1$ Expanding binomially, the $i$ th term of the above sum becomes $t_{ij}^4+ 4t_{ij}^3t_{ik} + 6 t_{ij}^2t_{ik}^2 + 4 t_{ij}t_{ik}^3+ t_{ik}^4=2$ Summing over $i$ and using (1), we obtain $\sum_i 4t_{ij}^3t_{ik} + 6 t_{ij}^2t_{ik}^2 + 4 t_{ij}t_{ik}^3 = 0, \hspace{10pt} (a)$ Similarly, taking $x_j= -x_k= \frac{1}{2^{1/4}}$ , we obtain $\sum_i -4t_{ij}^3t_{ik} + 6 t_{ij}^2t_{ik}^2 - 4 t_{ij}t_{ik}^3 = 0, \hspace{10pt} (b)$ Summing (a) and (b), we obtain $\sum_i t_{ij}^2t_{ik}^2 =0$ Since the matrix $T$ is real, we conclude $t_{ij}^2t_{ik}^2=0, \forall i$ . This implies that there is at most one non-zero element at each row of $T$ . On the other hand, due to the requirement of invertibility of $T$ , we need at least one non-zero element per row. This, along with the condition (1) implies that the set of all feasible $\mathcal{T}_N$ is the generalized permutation matrices , where the non-zero element $1$ could have either positive or negative sign. Hence, there are exactly $2^{N} N!$ of such linear transformations.

Discussion It can be easily seen that the proof easily extends beyond $p=4$ . Since in this case, the only feasible transformations are rather trivial (essentially shuffling and switching the direction of bases), we do not obtain any useful physical theory in this way, for any norm other than $p=1$ (Probability theory) and $p=2$ (Quantum Mechanics), the two main pillars of modern mathematics and physics.