Hilbert Spaces

I had the great opportunity of lecturing about the core mathematics, wherein I gave the lecture on Hilbert Spaces. Students who were shrewd enough and most interested in math were really astonished by the core content.

A Hilbert space is a central concept in both quantum mechanics and mathematical physics. It’s a complete vector space that comes equipped with an inner product, which allows for the measurement of lengths (norms) and angles (inner products) between vectors

Hilbert Spaces in Quantum Field Theory

In QFT, the states of the system are described by fields, which are also represented by vectors in infinite-dimensional Hilbert spaces. The space is typically constructed by using the Fock space, which is a specific kind of Hilbert space that allows for the creation and annihilation of particles (i.e., the number of particles is not fixed).

Hilbert Spaces in Quantum Mechanics

In quantum mechanics, the state of a system is represented by a vector in a Hilbert space. Key reasons for using Hilbert spaces include:

(a) Superposition Principle

Quantum states can be added together (superposition), which is naturally described by the vector space structure.

(b) Measurement and Observables

Observables (like position, momentum, or energy) are represented by self-adjoint operators on a Hilbert space. The eigenvalues of these operators correspond to possible measurement outcomes.

(c) Probability Interpretation

The probability of a measurement outcome is given by the square of the inner product

Geometric intuition plays an important role in many aspects of Hilbert space theory. Exact analogs of the pythogorean theorem and parallellogram law hold in a Hilbert space. At a deeper level, perpendicular projection onto a linear space plays a significant role in optimization problems and other aspects of the theory. An element of a Hilbert space can be uniquely specified by its coordinates with respect to an orthonormal basis, in analogy with cartesian coordinates in classical geometry. When this basis is countably infinite, it allows identifying the Hilbert space with the space of the infinite sequences that are square summable. The latter space is often in the older literature referred to as the Hilbert space.