Fields

From Sean Carroll's "Quanta and Fields"

Key Concepts

1. Fields as the Fabric of Reality

Fields, according to our best current understanding, are among the most fundamental ingredients of nature. In quantum field theory (QFT), particles emerge as excitations or quanta of underlying fields, rather than as independent entities moving through space.

Not Just in Spacetime—They Are Spacetime
Fields are not simply objects within spacetime; they constitute the dynamical degrees of freedom that fill all of spacetime. Efforts in quantum gravity suggest there may be something even deeper underlying these fields, but such ideas remain speculative.
Debates in Interpretation
Just as quantum mechanics spurs debates about measurement and reality, QFT raises profound questions about fields, vacuum energy, and the structure of spacetime. Theoretical work continues to refine our understanding.

2. QFT Basics and the Role of the Hamiltonian

In quantum theory, energy is introduced through the Hamiltonian, the operator that drives time evolution in the Schrödinger equation. In QFT, every field has its own Hamiltonian, which dictates how the field behaves and what energies are allowed.

Energy of a Field
A field’s Hamiltonian encodes both kinetic and potential energy terms. When the field is quantized, these translate into possible excitations of the field—what we call “particles.”
Interaction vs. Free Fields
Free Fields:
Fields that do not interact with other fields or with themselves. Their Hamiltonian typically looks like a sum of non-interacting modes—think of them as the simplest “no forces” scenario.
Interacting Fields:
In reality, fields often interact, introducing complexities like scattering, bound states, and nonlinear effects. These interactions make the theory rich—and much more challenging.

A common example for the Hamiltonian density $\mathcal{H}$ of a free scalar field $\phi(\mathbf{x}, t)$ is:

\[ \mathcal{H} \;=\; \frac{1}{2}\,\pi^2(\mathbf{x}, t) \;+\;\frac{1}{2}\,(\nabla \phi(\mathbf{x}, t))^2 \;+\;\frac{1}{2}\,m^2\,\phi^2(\mathbf{x}, t), \]

where $\pi(\mathbf{x}, t)$ is the conjugate momentum to $\phi(\mathbf{x}, t)$. Integrating this over all space gives the total Hamiltonian $H$.

3. Free Fields and Mass

Free fields have no interaction potential. They spread out over space instead of being localized to a particular region. Even if a free field is said to carry “mass,” it does not automatically localize the field itself. Instead, the mass term in the Hamiltonian affects how excitations (particles) of the field propagate or decay.

Why 'Mass' Still Matters
In QFT, “mass” determines the dispersion relation of excitations—how the energy relates to momentum:

$$ E^2 \;=\; \mathbf{p}^2c^2 \;+\; m^2c^4. $$

A massive particle’s energy depends differently on momentum than a massless particle’s (like a photon). Nevertheless, the field itself remains everywhere, with quanta that appear as localized excitations when observed.

4. Modes: The Building Blocks of Field Configurations

A powerful way to think about fields is by decomposing them into modes—wave-like solutions that span all of space.

Plane Waves
A free field can be represented as a superposition of plane waves with different wavevectors $\mathbf{k}$. Mathematically, one common expression is:

$$ \phi(\mathbf{x}, t) \;=\; \int \frac{d^3k}{(2\pi)^3} \,\Big[\, a_{\mathbf{k}}\,e^{\,i(\mathbf{k}\cdot\mathbf{x} \;-\; \omega_{\mathbf{k}}\,t)} \;+\; a_{\mathbf{k}}^*\,e^{-\,i(\mathbf{k}\cdot\mathbf{x} \;-\; \omega_{\mathbf{k}}\,t)} \Big], $$

where $\omega_{\mathbf{k}} = \sqrt{\mathbf{k}^2 + m^2}$ for a scalar field (in units where $c = 1$).

Fourier Transform and $\mathbf{k}$-Space
Going from real space $\mathbf{x}$ to momentum space $\mathbf{k}$ via the Fourier transform simplifies the mathematics of wave propagation and lays the groundwork for quantization.
Energy of a Mode
For a free scalar field, each $\mathbf{k}$-mode behaves like a harmonic oscillator with energy levels spaced by $\hbar \omega_{\mathbf{k}}$. This fact underlies the idea that a free quantum field is essentially an infinite collection of quantum harmonic oscillators.

5. Wavefunctions of Fields

When multiple particles or excitations are possible, we no longer have just a “wavefunction of one particle’s position.” Instead, we talk about a wavefunction over field configurations:

Configuration Space
The “position” variable in standard quantum mechanics is replaced by the entire configuration of the field at each point in space. Formally, you might write a wavefunctional $\Psi[\phi]$, which assigns amplitudes to every possible shape (configuration) of the field $\phi$.
No Single-Particle Restriction
This viewpoint naturally accommodates different particle numbers—one, two, or many excitations—without needing separate wavefunctions for each possible number of particles.

6. Particles from Fields (Preview)

While this chapter focuses on fields themselves, the notion of “particle” arises when a field is excited in quantized energy levels (the modes). QFT unifies the idea of wave-particle duality: both wave-like and particle-like pictures emerge from the same underlying field.

Creation and Annihilation Operators (Preview)
In a rigorous formulation, each mode $\mathbf{k}$ is treated as a quantized harmonic oscillator. We introduce operators $\hat{a}^\dagger_{\mathbf{k}}$ (creation) and $\hat{a}_{\mathbf{k}}$ (annihilation) that raise or lower the number of quanta in that mode. For example, you might see:

$$ \hat{\phi}(\mathbf{x}) \;=\; \int \frac{d^3k}{(2\pi)^3} \,\frac{1}{\sqrt{2\,\omega_{\mathbf{k}}}} \Big( \hat{a}{\mathbf{k}}\,e^{\,i\,\mathbf{k}\cdot \mathbf{x}} \;+\; \hat{a}{\mathbf{k}}^\dagger\,e^{-\,i\,\mathbf{k}\cdot \mathbf{x}} \Big). $$

We’ll see how these operators formalize the link between fields and particles in upcoming sections.

7. Fields of the World

Every known fundamental particle corresponds to a quantum field: electrons (electron field), photons (electromagnetic field), quarks (quark field), gluons (gluon field), and so on. Each field plays a role in the grand tapestry we call the Standard Model of particle physics.

Speculative Frontiers
The quest for quantum gravity and the nature of spacetime continues to drive research. Whether fields themselves are emergent from deeper structures—like strings, loops, or something else—remains an active area of exploration.

Takeaways

Fields Are Primary
In QFT, fields underpin the dynamics and properties of what we observe as “particles.” The concept of an isolated particle traveling through space is replaced by the concept of excitations in a field permeating the entire universe.
The Hamiltonian Defines Energy
The Hamiltonian for a field controls its evolution over time, just as in ordinary quantum mechanics. For free fields, it resembles an infinite collection of harmonic oscillators, each mode providing a quantized energy level.
Modes and Superposition
By decomposing fields into plane-wave modes, we reveal how excitations form and propagate. The Fourier transform is central to this picture, illustrating how fields can be built up from an infinite set of simpler wave components.
Wavefunctions Over Field Configurations
Instead of merely tracking the position of a single particle, QFT uses wavefunctionals that assign amplitudes to every conceivable field shape. This naturally incorporates situations with variable particle numbers.
Linking Fields and Particles
“Particles” are best seen as discrete excitations of underlying fields, with creation and annihilation operators formalizing their quantum behavior. Future sections will explore how interactions shape these excitations, including scattering processes and bound states.
Speculative Depths
While QFT is incredibly successful, many open questions remain—especially about gravity and spacetime. We continue to explore whether fields are truly fundamental or themselves emergent from an even deeper theory.

Resources

Quanta and Fields by Sean Carroll
Peskin & Schroeder, An Introduction to Quantum Field Theory
David Tong, Lectures on Quantum Field Theory

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