The Heisenberg Picture of Quantum Mechanics and Its Role in Quantum Field Theory

Patrick Reany

22 April 2024

Patrick:
How does the Heisenberg picture of quantum mechanics fit into quantum field theory?

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The Heisenberg picture of quantum mechanics provides an alternative formulation to the more familiar Schrodinger picture. In the Heisenberg picture, the operators (such as observables) incorporate a time dependency, while the state vectors remain time-independent.

Heisenberg Picture vs. Schrodinger Picture

• In the Schrodinger picture, the state vectors evolve in time, while the operators (such as position, momentum, and energy) remain constant.
• In contrast, the Heisenberg picture reverses this: the state vectors are fixed, and the operators evolve with time.
• The difference between the two pictures lies in the basis used for describing the system. The Heisenberg picture employs an arbitrary fixed basis, whereas the Schrodinger picture uses a basis that evolves with time.
• Mathematically, the Heisenberg picture is more natural for relativistic theories and maintains Lorentz invariance, as it treats time and space symmetrically.

Equivalence and Unitary Transformation

• The Heisenberg and Schrodinger pictures are equivalent. They differ only by a unitary transformation (a basis change) in Hilbert space.
• The Stone-von Neumann theorem establishes this equivalence, ensuring that both pictures yield the same physical predictions.
• The Heisenberg picture is particularly useful for relativistic quantum field theories, where Lorentz invariance plays a crucial role.

Quantum Field Theory and the Heisenberg Picture

Quantum field theory (QFT) extends quantum mechanics to describe fields (such as the electromagnetic field) and their interactions with particles.

In QFT, the Heisenberg picture remains applicable. Operators representing fields (like the field strength or particle creation/annihilation operators) evolve with time, while the state vectors remain fixed.

The Heisenberg picture allows us to express QFT in an arbitrary basis, even when the Hamiltonian (the operator representing the total energy) is not necessarily diagonal.

In summary, the Heisenberg picture provides an alternative perspective on quantum mechanics, emphasizing the time evolution of operators while keeping state vectors constant. Its application extends naturally to quantum field theory, where it plays a crucial role in describing fundamental interactions among particles and fields.