In the schrodinger wave formulation of quantum mechanics, analytic continuity of the wave function plays an important part. What replaces analytic continuity when one switches to the Matrix formulation of quantum mechanics?

Wave Continuity in Matrix Formulation of Quantum Mechanics?

In the matrix formulation of quantum mechanics (developed primarily by Heisenberg), the concept that replaces
analytic continuity of the wave function is the algebraic structure and consistency of operators in Hilbert space.
While the Schrödinger formulation relies on the analytic properties of the wave function, the matrix formulation emphasizes the following key features:

• Hermiticity and Self-Adjoint Operators:Physical observables in the matrix formulation are represented by Hermitian operators (self-adjoint matrices), ensuring real eigenvalues
  corresponding to measurable quantities. The algebra of these operators must be consistent and preserve the physical requirements of the theory.
• Commutation Relations:The fundamental relationships between observables, such as the canonical commutation relations
  [ ˆx, ˆp ] = iℏ, play a central role. These relations ensure the internal consistency of quantum mechanics
  and replace the need for analytic continuity of the wave function.
• Unitary Evolution:In the Schrödinger picture, the time evolution of the wave function must be continuous and differentiable, governed by the Schrödinger equation.
  In the matrix formulation, the time evolution is encoded in the unitary evolution of state vectors or density matrices in Hilbert space, satisfying
  ˆU(t)ˆU†(t) = ˆI.
• Spectrum and Eigenstates:The spectrum of the operators (eigenvalues) and their corresponding eigenstates provide the quantum mechanical predictions. These eigenvalues and
  eigenvectors are well-defined algebraically, independent of the notion of analytic continuity.
• Matrix Element Consistency:The elements of matrices in this formulation encode transition amplitudes between quantum states, and their consistency is guaranteed by the
  mathematical framework of linear algebra and Hilbert space theory.

In summary, while analytic continuity ensures the smoothness and well-defined behavior of the wave function in the Schrödinger picture, the matrix formulation
relies on the consistency of operator algebra, the structure of commutation relations, and the properties of Hilbert space. This shift reflects the abstract
algebraic nature of the matrix formulation, which avoids the explicit reliance on continuous functions.

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