Hilbert Space

Context

• In Quantum Mechanics the state of a physical system is represented by a vector in a Hilbert space: a complex vector space with an inner product.
• The term “Hilbert space” is often reserved for an infinite-dimensional inner product space having the property that it is complete or closed. However, the term is often used in a way that includes finite-dimensional spaces, which automatically satisfy the condition of completeness.
• We will use Dirac notation in which the vectors in the space are denoted by |v>, called a ket, where v is some symbol which identifies the vector.
• One could equally well use something like v. A multiple of a vector by a complex number c is written as c|v> -think of it as analogous to cv.
• In Dirac notation the inner product of the vectors |v> with |w> is written <v|w>. This resembles the ordinary dot product ~v · ~w except that one takes a complex conjugate of the vector on the left, thus think of ~v∗· ~w.

Problems

Quantum Theory

Unitary Gauge Transformations

• A reason to be skeptical about assigning a physical meaning directly to the Hilbert-space ingredients: quantum theory contains a little-known form of gauge invariance

so all these Hilbert-space ingredients are gauge-dependent, and their physical meaning is therefore highly suspect!

• Indeed, one can map any state-vector trajectory in the Hilbert space to any other state-vector trajectory
• A state-vector trajectory therefore contains no gauge-invariant physical content!
• What is gauge-invariant in textbook quantum theory?
• Measurement outcomes, measurement-outcome probabilities, and averages of the former over the latter, but now we are back to the only ingredients being these instrumentalist quantities, and se we are back to the category problem.

Solution

A New Foundation for Quantum Theory