# Lecture 3: Why Quantum Field Theory | Summary and Q&A

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February 12, 2024
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Lecture 3: Why Quantum Field Theory

## TL;DR

The free scalar field theory is quantized by promoting the classical equations of motion to quantum operator equations, finding the most general classical solution, promoting the integration constants to constant quantum operators, imposing canonical quantization conditions, and finding the commutation relations between the quantum operators.

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### Q: What are the steps involved in the quantization of the free scalar field theory?

The steps involved in the quantization of the free scalar field theory are: (1) promote the classical equations of motion to quantum operator equations, (2) find the most general classical solution, (3) promote the integration constants to constant quantum operators, and (4) impose canonical quantization conditions.

### Q: How is the classical solution of the free scalar field theory obtained?

The classical solution of the free scalar field theory is obtained by solving the classical equations of motion, resulting in a linear combination of plane waves. These plane waves form a basis for the classical solution.

### Q: How is the quantization of the free scalar field theory achieved?

The quantization of the free scalar field theory is achieved by promoting the classical solution to quantum operators, imposing canonical commutation relations between the creation and annihilation operators, and building the Hilbert space using these operators.

### Q: What are the commutation relations between the creation and annihilation operators in the free scalar field theory?

The commutation relations between the creation and annihilation operators in the free scalar field theory are given by [a_k, a_k'] = 0, [a_k, a_k†] = 2π^3 δ(k - k'), and [a_k†, a_k†'] = 0, where a_k and a_k† are the annihilation and creation operators labeled by the wave number k.

## Summary & Key Takeaways

• The free scalar field theory is quantized by applying the same steps as in the quantization of the harmonic oscillator: promoting classical equations of motion to quantum operator equations, finding the most general classical solution, promoting integration constants to constant quantum operators, and imposing canonical quantization conditions.

• The classical solution of the free scalar field theory is given by a linear combination of plane waves, where each plane wave corresponds to a basis solution.

• In quantum field theory, the basis solutions become quantum operators, and the commutation relations between the creation and annihilation operators are determined by dimensional analysis.

• The quantization of the free scalar field theory results in an infinite number of independent harmonic oscillators labeled by a continuous wave number.