bridge derivation: quantum mechanics to quantum field theory

[bhpayne]

• https://physics.stackexchange.com/questions/80824/what-is-the-difference-between-quantum-physics-quantum-theory-quantum-mechanic
• https://www.reddit.com/r/AskPhysics/comments/yfbt0b/quantum_mechanics_vs_quantum_field_theory/

[bhpayne]

prompt for Gemini 2.0 Flash Thinking Experimental on https://aistudio.google.com/prompts/new_chat

Quantum mechanics focuses on the behavior of individual particles at the atomic and subatomic level.

Quantum mechanics is the non-relativistic limit of Quantum Field Theory (QFT), a theory that combines Quantum Mechanics with Relativity.

QFT treats a particle as an excited state of an underlying physical field.

In mathematical Physics a derivation is comprised of steps and equations. Is there a derivation that bridges Quantum Mechanics with Quantum Field Theory? If yes, provide the steps of the derivation. Format the answer using Latex as a single file.

Latex is a software system for typesetting documents. LaTeX is widely used for the communication and publication of scientific documents and technical note-taking in many fields, owing partially to its support for complex mathematical notation.

Do not have equations inline with the text. Equations should be separate and inside a math environment for Latex:

\begin{equation}
\label{eq:}
\end{equation}

Each equation should have a left-hand side and right-hand side. Specify mathematical equations explicitly. Define the variables used in equations.

thoughts

The user is asking for a derivation that bridges Quantum Mechanics (QM) and Quantum Field Theory (QFT). The prompt also provides background information defining QM and QFT and their relationship. The prompt specifies the output format should be LaTeX, with equations displayed in a separate math environment and variables defined.

The core question is whether there is a derivation bridging QM to QFT. This is a tricky question because QFT is not strictly derived from QM in the sense of a mathematical deduction. QFT is a more fundamental and encompassing theory that reduces to QM in certain limits. It's more accurate to say that QFT generalizes QM to be consistent with special relativity and to handle particle creation and annihilation.

Therefore, a "derivation" in the strict mathematical sense might not exist. However, we can outline a conceptual and physical progression from QM to QFT, highlighting the key steps and concepts that motivate the transition. This would involve showing how the limitations of QM (incompatibility with relativity, fixed particle number) lead to the need for QFT and how QFT addresses these limitations.

Here's a possible thought process to structure the "derivation":

1. Start with the limitations of QM. QM is non-relativistic and doesn't naturally handle particle creation/annihilation. These are the primary motivations for moving to QFT.

2. Introduce the need for Relativity. High energies and velocities require relativity. The Klein-Gordon and Dirac equations are attempts to make QM relativistic, but they have issues (negative probabilities, single particle interpretation).

3. Shift to Fields. The concept of fields becomes central in QFT. Instead of particles, fields are the fundamental objects. Particles are excitations of these fields. This naturally incorporates particle creation and annihilation.

4. Canonical Quantization of Fields. Apply the quantization procedure to fields. Promote fields and their conjugate momenta to operators. Impose commutation/anti-commutation relations.

5. Particle Interpretation from Fields. Show how the quantized field operators can be interpreted in terms of creation and annihilation operators for particles. This bridges the field concept back to particle language, but in a more general way.

6. Vacuum and Multi-particle States. Introduce the vacuum state and how creation operators act on it to create multi-particle states.

7. Hamiltonian and Lagrangian Formalisms in QFT. Briefly mention how these formalisms are extended to fields.

8. Renormalization (brief mention). Acknowledge that QFT has infinities and renormalization is needed, although this is a very advanced topic and might be too detailed for a simple "derivation". Perhaps just mention that QFT requires techniques to handle infinities that are not present in QM.

9. Summarize the transition. Briefly reiterate the key shifts from QM to QFT: particles to fields, fixed particle number to variable particle number, non-relativistic to relativistic framework.

Now let's think about the Latex formatting and specific equations.

• Equations should be separate and labeled.
• Variables need to be defined.

We can use standard QM equations and build up towards the QFT concepts. For instance:

• Schrödinger Equation (QM): Start with this as the foundation of non-relativistic QM.
• Klein-Gordon Equation (Relativistic attempt, but flawed): Show an attempt to make QM relativistic, and point out its issues.
• Dirac Equation (Another relativistic attempt, better for spin-1/2, but still single particle): Show this for completeness, but still point out single particle limitations.
• Field Quantization: Show the expansion of a scalar field in terms of creation and annihilation operators. This is a key equation in QFT.
• Hamiltonian for a Free Scalar Field: Show the Hamiltonian in terms of creation and annihilation operators.

It's important to manage expectations. A full mathematical derivation of QFT from QM is not feasible in a simple step-by-step format. The goal is to present a conceptual progression and highlight the key mathematical and physical ideas that connect these theories. The "derivation" will be more of a motivational and conceptual bridge rather than a strict mathematical deduction.

Let's structure the LaTeX output now. Start with an introduction, then the limitations of QM, then the steps toward QFT, and finally a conclusion. Use equations with labels as requested. Define variables as they are introduced.

response

\documentclass{article}
\usepackage{amsmath}
\usepackage{amsfonts}
\usepackage{amssymb}

\title{Bridging Quantum Mechanics and Quantum Field Theory}
\date{}

\begin{document}
\maketitle

\section{Introduction}

Quantum Mechanics (QM) provides a highly successful description of physical phenomena at the atomic and subatomic level. It describes the behavior of matter and light in terms of quantization of energy, momentum, and other physical properties. However, standard Quantum Mechanics, as formulated in the early 20th century, has limitations when dealing with phenomena involving high energies and particle creation and annihilation.  Quantum Field Theory (QFT) emerges as a more fundamental framework that addresses these limitations by incorporating special relativity and treating particles as excitations of quantum fields. While a rigorous mathematical derivation of QFT directly from QM in the sense of deductive logic is not straightforward, we can outline a conceptual and physical progression, highlighting the key steps and modifications needed to transition from QM to QFT. This exposition will focus on the motivations and conceptual shifts rather than a strictly axiomatic derivation.

\section{Limitations of Quantum Mechanics}

Standard Quantum Mechanics, particularly its initial formulation based on the Schrödinger equation, is inherently non-relativistic.

\subsection{Non-Relativistic Nature}

The time-dependent Schrödinger equation treats time and space differently, which is inconsistent with the postulates of special relativity, where space and time are intertwined. The Schrödinger equation for a single particle in a potential $V(\mathbf{x})$ is given by:

\begin{equation}
\label{eq:Schrodinger}
i\hbar \frac{\partial}{\partial t} \Psi(\mathbf{x}, t) = \left( -\frac{\hbar^2}{2m} \nabla^2 + V(\mathbf{x}) \right) \Psi(\mathbf{x}, t)
\end{equation}
where:
\begin{itemize}
    \item $i$ is the imaginary unit.
    \item $\hbar$ is the reduced Planck constant.
    \item $\frac{\partial}{\partial t}$ is the partial derivative with respect to time $t$.
    \item $\Psi(\mathbf{x}, t)$ is the wave function of the particle, depending on position $\mathbf{x}$ and time $t$.
    \item $m$ is the mass of the particle.
    \item $\nabla^2 = \frac{\partial^2}{\partial x^2} + \frac{\partial^2}{\partial y^2} + \frac{\partial^2}{\partial z^2}$ is the Laplacian operator.
    \item $V(\mathbf{x})$ is the potential energy as a function of position $\mathbf{x}$.
\end{itemize}
This equation is first order in time and second order in spatial derivatives, reflecting its non-relativistic nature. Attempts to make QM relativistic by replacing the kinetic energy term with its relativistic counterpart in the Schrödinger equation lead to equations like the Klein-Gordon equation and the Dirac equation.

\subsection{Particle Number Problem}

In standard QM, the number of particles is fixed.  The wave function describes the probability amplitude for finding a fixed number of particles. Processes like particle creation and annihilation, which are common in high-energy physics and are naturally described by relativity (e.g., $E=mc^2$ allows for energy to convert into mass and vice versa), are not naturally incorporated within the framework of standard QM.

\section{Transition to Quantum Field Theory}

QFT addresses the limitations of QM by shifting the focus from particles to fields and incorporating special relativity from the outset. The transition involves several key steps.

\subsection{Relativistic Wave Equations as Field Equations}

Instead of interpreting relativistic wave equations like the Klein-Gordon equation and the Dirac equation as single-particle wave equations (which leads to issues like negative probabilities), we reinterpret them as classical field equations. For example, the Klein-Gordon equation becomes the equation of motion for a scalar field $\phi(x)$, where $x = (ct, \mathbf{x})$ is a spacetime four-vector:

\begin{equation}
\label{eq:KleinGordonClassical}
(\Box + m^2c^2/\hbar^2) \phi(x) = 0
\end{equation}
where:
\begin{itemize}
    \item $\Box = \partial^\mu \partial_\mu = \frac{1}{c^2}\frac{\partial^2}{\partial t^2} - \nabla^2$ is the d'Alembertian operator, with $\partial^\mu = (\frac{1}{c}\frac{\partial}{\partial t}, -\nabla)$ and $\partial_\mu = (\frac{1}{c}\frac{\partial}{\partial t}, \nabla)$.
    \item $m$ is the mass associated with the field.
    \item $c$ is the speed of light.
\end{itemize}
Similarly, the Dirac equation becomes the equation of motion for a Dirac field $\psi(x)$, a four-component spinor field.

\subsection{Canonical Quantization of Fields}

The crucial step in transitioning to QFT is to quantize these classical fields. We treat the fields as operators, just as we treat position and momentum as operators in QM.  We apply the procedure of canonical quantization to the fields and their conjugate momenta.

For a real scalar field $\phi(x)$, we define the conjugate momentum field $\pi(x) = \frac{\partial \mathcal{L}}{\partial (\partial_0 \phi(x))}$, where $\mathcal{L}$ is the Lagrangian density for the field. For the Klein-Gordon field, the Lagrangian density is:

\begin{equation}
\label{eq:LagrangianScalarField}
\mathcal{L} = \frac{1}{2} (\partial_\mu \phi(x)) (\partial^\mu \phi(x)) - \frac{1}{2} \frac{m^2c^2}{\hbar^2} \phi(x)^2
\end{equation}
The conjugate momentum is then $\pi(x) = \frac{1}{c^2} \frac{\partial \phi(x)}{\partial t}$.

We then promote the fields $\phi(x)$ and $\pi(x)$ to operators $\hat{\phi}(x)$ and $\hat{\pi}(x)$ and impose canonical commutation relations at equal times:

\begin{equation}
\label{eq:CommutationRelations}
[\hat{\phi}(\mathbf{x}, t), \hat{\pi}(\mathbf{y}, t)] = i\hbar \delta^{(3)}(\mathbf{x} - \mathbf{y})
\end{equation}
\begin{equation}
\label{eq:CommutationRelationsPhi}
[\hat{\phi}(\mathbf{x}, t), \hat{\phi}(\mathbf{y}, t)] = 0
\end{equation}
\begin{equation}
\label{eq:CommutationRelationsPi}
[\hat{\pi}(\mathbf{x}, t), \hat{\pi}(\mathbf{y}, t)] = 0
\end{equation}
where $\delta^{(3)}(\mathbf{x} - \mathbf{y})$ is the three-dimensional Dirac delta function.

\subsection{Mode Expansion and Particle Interpretation}

We can expand the field operator $\hat{\phi}(x)$ in terms of normal modes, which are solutions to the classical field equation (e.g., Klein-Gordon equation). For a real scalar field, the mode expansion in momentum space is:

\begin{equation}
\label{eq:ModeExpansionScalarField}
\hat{\phi}(x) = \int \frac{d^3\mathbf{p}}{(2\pi)^3} \frac{1}{\sqrt{2E_{\mathbf{p}}}} \left( \hat{a}_{\mathbf{p}} e^{-ip \cdot x/\hbar} + \hat{a}^\dagger_{\mathbf{p}} e^{ip \cdot x/\hbar} \right)
\end{equation}
where:
\begin{itemize}
    \item $\mathbf{p}$ is the momentum.
    \item $E_{\mathbf{p}} = \sqrt{|\mathbf{p}|^2c^2 + m^2c^4}$ is the relativistic energy corresponding to momentum $\mathbf{p}$.
    \item $p \cdot x = E_{\mathbf{p}}t - \mathbf{p} \cdot \mathbf{x}$ is the four-vector dot product.
    \item $\hat{a}_{\mathbf{p}}$ and $\hat{a}^\dagger_{\mathbf{p}}$ are annihilation and creation operators, respectively.
\end{itemize}
From the commutation relations of the fields, we can deduce the commutation relations for the creation and annihilation operators:

\begin{equation}
\label{eq:CreationAnnihilationCommutation}
[\hat{a}_{\mathbf{p}}, \hat{a}^\dagger_{\mathbf{q}}] = (2\pi)^3 \delta^{(3)}(\mathbf{p} - \mathbf{q})
\end{equation}
\begin{equation}
\label{eq:AnnihilationAnnihilationCommutation}
[\hat{a}_{\mathbf{p}}, \hat{a}_{\mathbf{q}}] = 0
\end{equation}
\begin{equation}
\label{eq:CreationCreationCommutation}
[\hat{a}^\dagger_{\mathbf{p}}, \hat{a}^\dagger_{\mathbf{q}}] = 0
\end{equation}
These are the same commutation relations as for the creation and annihilation operators of a quantum harmonic oscillator, and they indicate that $\hat{a}^\dagger_{\mathbf{p}}$ creates a particle with momentum $\mathbf{p}$ and energy $E_{\mathbf{p}}$, while $\hat{a}_{\mathbf{p}}$ annihilates such a particle.

\subsection{Vacuum State and Multi-Particle States}

We define a vacuum state $|0\rangle$ which is annihilated by all annihilation operators:

\begin{equation}
\label{eq:VacuumAnnihilation}
\hat{a}_{\mathbf{p}} |0\rangle = 0 \quad \forall \mathbf{p}
\end{equation}
Multi-particle states are created by applying creation operators to the vacuum state. For example, a one-particle state with momentum $\mathbf{p}$ is given by $\hat{a}^\dagger_{\mathbf{p}} |0\rangle$, and a two-particle state with momenta $\mathbf{p}_1$ and $\mathbf{p}_2$ is given by $\hat{a}^\dagger_{\mathbf{p}_1} \hat{a}^\dagger_{\mathbf{p}_2} |0\rangle$. This formalism naturally allows for the description of states with varying numbers of particles.

\subsection{Hamiltonian in QFT}

The Hamiltonian operator in QFT, obtained from the Lagrangian density, can be expressed in terms of creation and annihilation operators. For the real scalar field, the Hamiltonian is:

\begin{equation}
\label{eq:HamiltonianScalarField}
\hat{H} = \int \frac{d^3\mathbf{p}}{(2\pi)^3} E_{\mathbf{p}} \hat{a}^\dagger_{\mathbf{p}} \hat{a}_{\mathbf{p}} + E_0
\end{equation}
where $E_0$ is an infinite constant vacuum energy that often needs to be handled through normal ordering or other regularization techniques.  The term $\hat{a}^\dagger_{\mathbf{p}} \hat{a}_{\mathbf{p}}$ is the number operator for particles with momentum $\mathbf{p}$, and $E_{\mathbf{p}} \hat{a}^\dagger_{\mathbf{p}} \hat{a}_{\mathbf{p}}$ represents the energy of these particles.  This Hamiltonian describes a system of non-interacting relativistic particles. Interactions are introduced by adding interaction terms to the Lagrangian density, leading to more complex QFTs.

\section{Conclusion}

The transition from Quantum Mechanics to Quantum Field Theory is not a strict mathematical derivation but rather a conceptual and physical generalization motivated by the limitations of QM in relativistic regimes and its inability to describe particle creation and annihilation.  QFT overcomes these limitations by:

\begin{itemize}
    \item Shifting the fundamental objects from particles to fields.
    \item Quantizing these fields using canonical quantization, promoting them to operators.
    \item Interpreting particles as quanta of these fields, i.e., excitations of the field modes.
    \item Naturally incorporating special relativity and allowing for a variable number of particles.
\end{itemize}
While this exposition provides a conceptual bridge and outlines the key steps and equations, a full mathematical derivation of QFT from QM as a limiting case is a complex topic involving renormalization, path integrals, and advanced mathematical techniques.  However, the conceptual framework described here elucidates the fundamental shift in perspective and the key physical ideas that distinguish QFT from QM and make it a more comprehensive theory of nature at the fundamental level.

\end{document}

[bhpayne]

Manual minor improvements

\documentclass{article}
\usepackage{amsmath}
\usepackage{amsfonts} % for \Box

% margins of 1 inch:
\setlength{\topmargin}{-.5in}
\setlength{\textheight}{9in}
\setlength{\oddsidemargin}{0in}
\setlength{\textwidth}{6.5in}

\usepackage[pdftex]{hyperref} % hyperlink equation and bibliographic citations

\title{Bridging Quantum Mechanics and Quantum Field Theory}
\date{\today}
\author{Ben Payne with Gemini 2.0}

\begin{document}
\maketitle

\section{Introduction}

Quantum Mechanics (QM) provides a highly successful description of physical phenomena at the atomic and subatomic level. It describes the behavior of matter and light in terms of quantization of energy, momentum, and other physical properties. However, standard Quantum Mechanics, as formulated in the early 20th century, has limitations when dealing with phenomena involving high energies and particle creation and annihilation.  Quantum Field Theory (QFT) emerges as a more fundamental framework that addresses these limitations by incorporating special relativity and treating particles as excitations of quantum fields. While a rigorous mathematical derivation of QFT directly from QM in the sense of deductive logic is not straightforward, we can outline a conceptual and physical progression, highlighting the key steps and modifications needed to transition from QM to QFT. This exposition will focus on the motivations and conceptual shifts rather than a strictly axiomatic derivation.

\section{Limitations of Quantum Mechanics}

Standard Quantum Mechanics, particularly its initial formulation based on the Schrödinger equation, is inherently non-relativistic.

\subsection{Non-Relativistic Nature}

The time-dependent Schrödinger equation treats time and space differently, which is inconsistent with the postulates of special relativity, where space and time are intertwined. The Schrödinger equation for a single particle in a potential $V(\mathbf{x})$ is given by:

\begin{equation}
\label{eq:Schrodinger}
i\hbar \frac{\partial}{\partial t} \Psi(\mathbf{x}, t) = \left( -\frac{\hbar^2}{2m} \nabla^2 + V(\mathbf{x}) \right) \Psi(\mathbf{x}, t)
\end{equation}
where:
\begin{itemize}
    \item $i$ is the imaginary unit.
    \item $\hbar$ is the reduced Planck constant.
    \item $\frac{\partial}{\partial t}$ is the partial derivative with respect to time $t$.
    \item $\Psi(\mathbf{x}, t)$ is the wave function of the particle, depending on position $\mathbf{x}$ and time $t$.
    \item $m$ is the mass of the particle.
    \item $V(\mathbf{x})$ is the potential energy as a function of position $\mathbf{x}$.
\end{itemize}
and
\begin{equation}
    \nabla^2 = \frac{\partial^2}{\partial x^2} + \frac{\partial^2}{\partial y^2} + \frac{\partial^2}{\partial z^2}
\end{equation}
is the Laplacian operator.

Eq.~\ref{eq:Schrodinger} is first order in time and second order in spatial derivatives, reflecting its non-relativistic nature. Attempts to make QM relativistic by replacing the kinetic energy term with its relativistic counterpart in the Schrödinger equation lead to equations like the Klein-Gordon equation and the Dirac equation.

\subsection{Particle Number Problem}

In standard QM, the number of particles is fixed.  The wave function describes the probability amplitude for finding a fixed number of particles. Processes like particle creation and annihilation, which are common in high-energy physics and are naturally described by relativity (e.g., $E=mc^2$ allows for energy to convert into mass and vice versa), are not naturally incorporated within the framework of standard QM.

\section{Transition to Quantum Field Theory}

QFT addresses the limitations of QM by shifting the focus from particles to fields and incorporating special relativity from the outset. The transition involves several key steps.

\subsection{Relativistic Wave Equations as Field Equations}

Instead of interpreting relativistic wave equations like the Klein-Gordon equation and the Dirac equation as single-particle wave equations (which leads to issues like negative probabilities), we reinterpret them as classical field equations. For example, the Klein-Gordon equation becomes the equation of motion for a scalar field $\phi(x)$, where $x = (ct, \mathbf{x})$ is a spacetime four-vector:

\begin{equation}
\label{eq:KleinGordonClassical}
(\Box + m^2c^2/\hbar^2) \phi(x) = 0
\end{equation}
where:
\begin{itemize}
    \item $m$ is the mass associated with the field.
    \item $c$ is the speed of light.
\end{itemize}
and
\begin{equation}
    \Box = \partial^\mu \partial_\mu = \frac{1}{c^2}\frac{\partial^2}{\partial t^2} - \nabla^2
\end{equation}
is the d'Alembertian operator, with
\begin{equation}
    \partial^\mu = (\frac{1}{c}\frac{\partial}{\partial t}, -\nabla)
\end{equation}
and
\begin{equation}
    \partial_\mu = (\frac{1}{c}\frac{\partial}{\partial t}, \nabla)
\end{equation}

Similarly, the Dirac equation becomes the equation of motion for a Dirac field $\psi(x)$, a four-component spinor field.

\subsection{Canonical Quantization of Fields}

The crucial step in transitioning to QFT is to quantize these classical fields. We treat the fields as operators, just as we treat position and momentum as operators in QM.  We apply the procedure of canonical quantization to the fields and their conjugate momenta.

For a real scalar field $\phi(x)$, we define the conjugate momentum field 
\begin{equation}
    \pi(x) = \frac{\partial \mathcal{L}}{\partial (\partial_0 \phi(x))}
\end{equation}
 where $\mathcal{L}$ is the Lagrangian density for the field. For the Klein-Gordon field, the Lagrangian density is:

\begin{equation}
\label{eq:LagrangianScalarField}
\mathcal{L} = \frac{1}{2} (\partial_\mu \phi(x)) (\partial^\mu \phi(x)) - \frac{1}{2} \frac{m^2c^2}{\hbar^2} \phi(x)^2
\end{equation}
The conjugate momentum is then 
\begin{equation}
\pi(x) = \frac{1}{c^2} \frac{\partial \phi(x)}{\partial t}    
\end{equation}


We then promote the fields $\phi(x)$ and $\pi(x)$ to operators $\hat{\phi}(x)$ and $\hat{\pi}(x)$ and impose canonical commutation relations at equal times:

\begin{equation}
\label{eq:CommutationRelations}
[\hat{\phi}(\mathbf{x}, t), \hat{\pi}(\mathbf{y}, t)] = i\hbar \delta^{(3)}(\mathbf{x} - \mathbf{y})
\end{equation}
\begin{equation}
\label{eq:CommutationRelationsPhi}
[\hat{\phi}(\mathbf{x}, t), \hat{\phi}(\mathbf{y}, t)] = 0
\end{equation}
\begin{equation}
\label{eq:CommutationRelationsPi}
[\hat{\pi}(\mathbf{x}, t), \hat{\pi}(\mathbf{y}, t)] = 0
\end{equation}
where $\delta^{(3)}(\mathbf{x} - \mathbf{y})$ is the three-dimensional Dirac delta function.

\subsection{Mode Expansion and Particle Interpretation}

We can expand the field operator $\hat{\phi}(x)$ in terms of normal modes, which are solutions to the classical field equation (e.g., Klein-Gordon equation). For a real scalar field, the mode expansion in momentum space is:

\begin{equation}
\label{eq:ModeExpansionScalarField}
\hat{\phi}(x) = \int \frac{d^3\mathbf{p}}{(2\pi)^3} \frac{1}{\sqrt{2E_{\mathbf{p}}}} \left( \hat{a}_{\mathbf{p}} e^{-ip \cdot x/\hbar} + \hat{a}^\dagger_{\mathbf{p}} e^{ip \cdot x/\hbar} \right)
\end{equation}
where:
\begin{itemize}
    \item $\mathbf{p}$ is the momentum.
    \item $E_{\mathbf{p}} = \sqrt{|\mathbf{p}|^2c^2 + m^2c^4}$ is the relativistic energy corresponding to momentum $\mathbf{p}$.
    \item $p \cdot x = E_{\mathbf{p}}t - \mathbf{p} \cdot \mathbf{x}$ is the four-vector dot product.
    \item $\hat{a}_{\mathbf{p}}$ and $\hat{a}^\dagger_{\mathbf{p}}$ are annihilation and creation operators, respectively.
\end{itemize}
From the commutation relations of the fields, we can deduce the commutation relations for the creation and annihilation operators:

\begin{equation}
\label{eq:CreationAnnihilationCommutation}
[\hat{a}_{\mathbf{p}}, \hat{a}^\dagger_{\mathbf{q}}] = (2\pi)^3 \delta^{(3)}(\mathbf{p} - \mathbf{q})
\end{equation}
\begin{equation}
\label{eq:AnnihilationAnnihilationCommutation}
[\hat{a}_{\mathbf{p}}, \hat{a}_{\mathbf{q}}] = 0
\end{equation}
\begin{equation}
\label{eq:CreationCreationCommutation}
[\hat{a}^\dagger_{\mathbf{p}}, \hat{a}^\dagger_{\mathbf{q}}] = 0
\end{equation}
These are the same commutation relations as for the creation and annihilation operators of a quantum harmonic oscillator, and they indicate that $\hat{a}^\dagger_{\mathbf{p}}$ creates a particle with momentum $\mathbf{p}$ and energy $E_{\mathbf{p}}$, while $\hat{a}_{\mathbf{p}}$ annihilates such a particle.

\subsection{Vacuum State and Multi-Particle States}

We define a vacuum state $|0\rangle$ which is annihilated by all annihilation operators:

\begin{equation}
\label{eq:VacuumAnnihilation}
\hat{a}_{\mathbf{p}} |0\rangle = 0 \quad \forall \mathbf{p}
\end{equation}
Multi-particle states are created by applying creation operators to the vacuum state. For example, a one-particle state with momentum $\mathbf{p}$ is given by $\hat{a}^\dagger_{\mathbf{p}} |0\rangle$, and a two-particle state with momenta $\mathbf{p}_1$ and $\mathbf{p}_2$ is given by $\hat{a}^\dagger_{\mathbf{p}_1} \hat{a}^\dagger_{\mathbf{p}_2} |0\rangle$. This formalism naturally allows for the description of states with varying numbers of particles.

\subsection{Hamiltonian in QFT}

The Hamiltonian operator in QFT, obtained from the Lagrangian density, can be expressed in terms of creation and annihilation operators. For the real scalar field, the Hamiltonian is:

\begin{equation}
\label{eq:HamiltonianScalarField}
\hat{H} = \int \frac{d^3\mathbf{p}}{(2\pi)^3} E_{\mathbf{p}} \hat{a}^\dagger_{\mathbf{p}} \hat{a}_{\mathbf{p}} + E_0
\end{equation}
where $E_0$ is an infinite constant vacuum energy that often needs to be handled through normal ordering or other regularization techniques.  The term $\hat{a}^\dagger_{\mathbf{p}} \hat{a}_{\mathbf{p}}$ is the number operator for particles with momentum $\mathbf{p}$, and $E_{\mathbf{p}} \hat{a}^\dagger_{\mathbf{p}} \hat{a}_{\mathbf{p}}$ represents the energy of these particles.  This Hamiltonian describes a system of non-interacting relativistic particles. Interactions are introduced by adding interaction terms to the Lagrangian density, leading to more complex QFTs.

\section{Conclusion}

The transition from Quantum Mechanics to Quantum Field Theory is not a strict mathematical derivation but rather a conceptual and physical generalization motivated by the limitations of QM in relativistic regimes and its inability to describe particle creation and annihilation.  QFT overcomes these limitations by:

\begin{itemize}
    \item Shifting the fundamental objects from particles to fields.
    \item Quantizing these fields using canonical quantization, promoting them to operators.
    \item Interpreting particles as quanta of these fields, i.e., excitations of the field modes.
    \item Naturally incorporating special relativity and allowing for a variable number of particles.
\end{itemize}
While this exposition provides a conceptual bridge and outlines the key steps and equations, a full mathematical derivation of QFT from QM as a limiting case is a complex topic involving renormalization, path integrals, and advanced mathematical techniques.  However, the conceptual framework described here elucidates the fundamental shift in perspective and the key physical ideas that distinguish QFT from QM and make it a more comprehensive theory of nature at the fundamental level.

\end{document}