Classical Treatment of Quantum Field - Quantizing Schrodinger Field

1. Classical Treatment of Quantum Field:

   • Treat $�$ as a classical field obeying the Schrödinger equation.
   • The Schrödinger equation is treated as a classical equation of motion.

2. Action of the System:

   • Constructed the action $�$ for the system, which leads to the Schrödinger equation as the equations of motion.
   • The action is given by $�=\int ��\int {�}^3�\mathcal{�}$, where $\mathcal{�}={�}^{\ast }(�\mathrm{\hslash }{\mathrm{\partial }}_{�}-\widehat{�})�$.

3. Lagrangian Density and Lagrangian:

   • Introduced the Lagrangian density within square brackets.
   • The Lagrangian is obtained by integrating the Lagrangian density over all of space.
   • Emphasized that the action is a functional of $�$ since it depends on the entire specification of $�$ across space and time.

4. Quantization:

   • Decomposed $�$ into a sum over wave functions ${�}_{�}$ of the Hamiltonian.
   • Expressed Lagrangian and Hamiltonian in terms of dynamical variables ${�}_{�}$ and ${�}_{�}^{\ast }$.
   • Established the commutation relation $[{�}_{�},{�}_{�}^{\ast }]={�}_{��}$.

5. Quantum Hamiltonian:

   • Constructed the quantum Hamiltonian for the system as $�={\sum }_{�}{�}_{�}{�}_{�}^{\ast }{�}_{�}$.
   • Recognized that the system is a collection of an infinite number of uncoupled harmonic oscillators.

6. Free Quantum Field Theory:

   • Emphasized that the resulting quantum mechanical system is a free theory.
   • The system consists of an infinite number of harmonic oscillators without interactions.

The discussion sets the groundwork for understanding a quantum field theory with free fields. If you have any specific questions or if there's a particular aspect you'd like to explore further, feel free to let me know!

Classical Treatment and Action:

1. Classical Field and Equation of Motion:
   • $�$ is treated as a classical field.
   • Schrödinger equation is treated classically: $�\mathrm{\hslash }\frac{\mathrm{\partial }�}{\mathrm{\partial }�}=-\frac{{\mathrm{\hslash }}^2}{2�}{\mathrm{\nabla }}^2�+��$.
2. Action for the System:
   • The action $�$ is given by $�=\int ��\int {�}^3�\mathcal{�}$.
   • The Lagrangian density $\mathcal{�}$ is constructed such that it yields the Schrödinger equation as the equation of motion.

Lagrangian and Hamiltonian Formulation:

3. Lagrangian Density and Lagrangian:

   • Lagrangian density $\mathcal{�}={�}^{\ast }(�\mathrm{\hslash }{\mathrm{\partial }}_{�}-\widehat{�})�$.
   • Lagrangian $�=\int {�}^3�\mathcal{�}$.

4. Field Decomposition and Dynamical Variables:

   • Decomposed $�$ into a sum over wave functions ${�}_{�}$ of the Hamiltonian: $�={\sum }_{�}{�}_{�}{�}_{�}$.
   • Introduced ${�}_{�}$ and ${�}_{�}^{\ast }$ as dynamical variables.

Quantization:

5. Quantum Field and Commutation Relation:

   • Promoted $�$ to an operator, ${�}_{�}$, and ${�}_{�}^{\ast }$ to its hermitian conjugate.
   • Imposed the commutation relation $[{�}_{�},{�}_{�}^{\ast }]={�}_{��}$.

6. Quantum Hamiltonian:

   • Constructed the quantum Hamiltonian: $�={\sum }_{�}{�}_{�}{�}_{�}^{\ast }{�}_{�}$.
   • ${�}_{�}$ represents the eigenvalues of the Hamiltonian operator.

Application and Free Field Theory:

7. Quantum Field System:

   • Described the resulting quantum system as an infinite collection of uncoupled harmonic oscillators.
   • The Hamiltonian represents a sum of terms for each harmonic oscillator.

8. Free Quantum Field Theory:

   • Emphasized that there is no interaction between different oscillators.
   • The system is a free theory despite originating from the Schrödinger equation with a potential term.

Equations and Implications:

9. Equations Recap:

   • Quantum Hamiltonian: $�={\sum }_{�}{�}_{�}{�}_{�}^{\ast }{�}_{�}$.
   • Commutation Relation: $[{�}_{�},{�}_{�}^{\ast }]={�}_{��}$.

10. Implications and Further Study:

    • Highlighted that this is a foundational step towards understanding free quantum field theories.
    • The system is a collection of independent harmonic oscillators, paving the way for more complex quantum field theories.