Standing electromagnetic waves quantization
In my recent research, I have focused on exploring the relationship between the field potential, the electric field, and the complex variable \alpha(t) within the context of quantum electrodynamics (QED). This post aims to provide an understanding of how these elements interact and lead to a deeper insight into the quantum properties of fields.
Canonical conjugate variables and quantization
At the outset, defining \alpha(t) and its conjugate \bar{\alpha}(t) is crucial:
\begin{aligned} & \alpha(t) = \frac{1}{2 \mathscr{E}^{(1)} _{sw}}(\omega \mathbf{A}(t) - i\mathbf{E}(t)) \\ & \bar{\alpha}(t) = \frac{1}{2 \mathscr{E}^{(1)} _{sw}}(\omega \mathbf{A}(t) + i\mathbf{E}(t)) \end{aligned}
These definitions facilitate expressing the vector potential \mathbf{A}(t) and the electric field \mathbf{E}(t) as functions of \alpha(t) and \bar{\alpha}(t), showcasing their relation to physical observables in a quantum field.
Transforming to position and momentum
Transitioning from field variables to mechanical analogs, position Q and momentum P, provides a classical perspective on quantum phenomena:
\begin{aligned} & Q = \Re{(\alpha_\lambda)} = \sqrt{\frac{\hbar}{2}} \frac{1}{\mathscr{E}^{(1)} _{sw}} \omega \mathbf{A}(t) \\ & P = \Im{(\alpha_\lambda)} = -\sqrt{\frac{\hbar}{2}} \frac{1}{\mathscr{E}^{(1)} _{sw}} \mathbf{E}(t) \end{aligned}
Here, Q and P reflect the real and imaginary parts of \alpha, respectively, analogous to the classical position and momentum in a harmonic oscillator, thus leading us to the quantization of the field.
Quantized Hamiltonian
With quantization, the Hamiltonian is formulated as:
\mathcal{H} = \frac{\omega}{2}(Q^2 + P^2)
This representation as a harmonic oscillator is not just a mathematical convenience but a fundamental aspect of quantum field theory that allows for the application of well-established quantum mechanics techniques.
Operators and commutation
To fully engage with quantum mechanics, I introduce the operators \mathbf{Q} and \mathbf{P} and their canonical commutation relation:
[\mathbf{Q}, \mathbf{P}] = i\hbar
This step is vital for ensuring that the quantum field theory respects the underlying principles of quantum mechanics.
Field and traveling waves
Building on the operator formalism, I explore the representation of the electric field in terms of traveling waves:
\mathbf{E}(x) = i\mathscr{E}^{(1)}_{\mathbf{k}} \mathbf{e} (\mathbf{b} _{\mathbf{k}} e^{i kx} - \mathbf{b} _{\mathbf{k}}^\dagger e^{-ikx})
This equation shows how the electric field can be expressed as a superposition of forward and backward traveling waves, encapsulating the wave-like nature of quantum fields.
Conclusion
My exploration of these concepts in QED is an ongoing journey, aimed at uncovering new aspects of how quantum fields behave and interact. By treating fields as quantum mechanical entities, I am able to apply the rigorous formalism of quantum mechanics to obtain deeper insights into the structure and dynamics of fields.
For more insights into this topic, you can find the details here.