Operators in quantum mechanics and quantum field theory are indeed functions, but they represent a crucial departure from classical physics and classical field theories. The "big deal" about operators in these theories lies in their underlying mathematical structure and the physical interpretation associated with them. Let me explain why operators are fundamental and different in quantum mechanics and quantum field theory compared to classical physics.
Mathematical structure: In quantum mechanics, the state of a system is described by a vector in a complex vector space called a Hilbert space. Operators in quantum mechanics are linear transformations on this vector space. They act on the state vectors to produce new state vectors or to extract physical observables.
Non-commutativity: One of the key distinctions between classical physics and quantum mechanics is the non-commutative nature of operators. In general, quantum mechanical operators do not commute, meaning the order in which operators are applied matters. This non-commutativity is a consequence of the uncertainty principle and has profound implications for the measurement process and the behavior of quantum systems.
Measurement and observables: In quantum mechanics, observables such as position, momentum, and energy are represented by self-adjoint operators. When a measurement is made on a quantum system, the system "collapses" into one of the eigenstates of the corresponding operator, yielding a definite value for the observable. The eigenvalues of these operators correspond to the possible outcomes of measurements.
Quantum superposition: Unlike classical physics, quantum mechanics allows for the existence of superposition states. Quantum mechanical operators can induce transitions between different superposition states, leading to phenomena such as interference and entanglement. These effects have no classical analog and are fundamental to the quantum mechanical description of the physical world.
Quantum field theory: In quantum field theory (QFT), operators play a crucial role in describing the fields and their dynamics. Fields are promoted to operators that act on a Fock space, which describes the quantum states of a system with a variable number of particles. The interactions and the evolution of fields are described by operator equations, such as the Klein-Gordon equation or the Dirac equation.
In summary, while operators in quantum mechanics and quantum field theory share some similarities with functions in classical physics, their non-commutative nature, the role they play in measurement and observables, and their connection to superposition and quantum field dynamics make them fundamentally different and crucial for understanding the quantum world.