How exactly do we know how should transformations of vectors of Hilbert space look like?

Question

There are transformations on physical states which induce unitary transformations of vectors in Hilbert space that correspond to these physical states. We demand that operators in Hilbert space be unitary and from this we mathematically deduce what should the algebra look like and what form should these transformations have in Hilbert space. These transformations are known as symmetry transformations for they preserve inner product or the norm, which is probability in QM.

So how can we be sure that our reasoning is correct just on these wage and weak arguments? Of course we can derive transformations and of course it all works, but it seems to me that this reasoning lacks something strong. Am I wrong??

Answer 1

The quantum theory has a long history, and reaches its first maturity with the papers of Heisenberg and Shroedinger which present their equations which show how a quantum system evolves with time.

Though their approaches are different, their results are equivalent, and the mathematicians (e.g., John Von Neumann) soon showed that one could present quantum mechanics as a linear theory based on the very standard mathematics of linear algebra on a complex field.

In this system, which is still used today, there are a very few axioms. One is that the system evolves following a unitary operator which is derived from the Hamiltonian operator which describes the system. Experimentalists then further manipulate the system with other unitary operators (mirrors, waveplates, etc), which preserve the total probability = 1, until they perform a measurement, usually via a projection operator (e.g., a beam splitter plus a detector), though there are more subtle methods.

This is all very simple once one begins to think about it in the "standard way", and the experimental results follow the theory very closely; in some cases, for Quantum Electrodynamics, to 14 decimal places or more.

To the beginning student it may seem vague, but with continued study and hard work it becomes clearer.