The trace of the identity operator is infinite in an infinite-dimensional separable Hilbert space, preventing the definition of a correctly normalized a priori probability for measurement outcomes.
A mathematical space used in quantum mechanics that has infinitely many dimensions (directions) and can be precisely described using a countable list of basic building blocks—similar to how you can build any color by mixing a specific set of primary colors.
normalized probability(in probability and quantum mechanics)
A properly scaled probability distribution where all possible outcomes add up to exactly 100% (or 1 in mathematical terms), so the probabilities make sense.
trace (of an operator)(in quantum mechanics and linear algebra)
A mathematical value you get by adding up certain numbers associated with an operator—think of it like a single summary number that tells you something important about how the operator works.
In the introduction to the first paper in the series of four entitled “On Rings of Operators”, Murray and von Neumann list two reasons why they are dissatisfied with the separable Hilbert space formulation of quantum mechanics. One has to do with a property of the trace operation, which is the operation appearing in the definition of the probabilities for measurement results (the Born rule), and the other with domain problems that arise for unbounded observable operators. The trace of the identi