50 Essential Theoretical Physics Questions Every Graduate Student Must Master [Part 1 of 3]

From Noether to Entropy to AdS/CFT...

Theoretical physics links math, experiment, and is inspired by fundamental physics (hopefully). These are fifty (nifty) Q&As to refresh core results, proofs, and tricks. Also, to just have some fun.

A solid grasp of these primary ideas is needed to understand classical mechanics, relativity, quantum theory, and whatever is beyond that. This list is for graduate-level candidates. Use it to find your weak spots or cement your knowledge.

Subscribe to this Substack for more theoretical physics insights (and the rest of the parts). I’ll also be covering the philosophy of language, philosophy of mind, condensed matter physics, and philosophy of physics in similar lists over the coming months.

Question 1: What’s the principle of least action and its significance?

A system’s path between two times extremizes the action, S. The action is the time integral of the Lagrangian, L = T − V.

T is kinetic energy; V is potential energy. The principle states the action’s variation is zero, so:

\(\delta S = \delta \int_{t_1}^{t_2} L(q, \dot{q}, t) ,dt = 0\)

This replaces Newton’s vector laws from the first year with a single scalar idea. You can derive the equations of motion from it. This variational method is general and is present in virtually any field theory.

Question 2: What’s Noether’s theorem and its implications?

Noether’s theorem connects continuous symmetries to conserved quantities. If a system’s action (e.g., the above) has a continuous symmetry, then a conserved charge exists.

• Time translation invariance produces energy conservation.

• Spatial translation invariance produces momentum conservation.

• Rotational invariance produces angular momentum conservation.

For field theory, we have a conserved current:

\(\partial_\mu J^\mu = 0, \quad J^\mu = \dfrac{\partial\mathcal{L}}{\partial(\partial_\mu\phi_a)} \delta\phi_a \)

If you have finite degrees of freedom, then the conserved quantity is:

\( \frac{d}{dt}\Bigl(\sum_i p_i \delta q_i\Bigr) = 0, \quad p_i = \frac{\partial L}{\partial \dot{q}_i} \)

Question 3: Define the Hamiltonian and derive Hamilton’s equations from the Lagrangian formalism.

The Hamiltonian H is the Legendre transform of the Lagrangian L. It’s given by:

\(H(q, p, t) = \sum_i p_i \dot{q}_i - L(q, \dot{q}, t)\)

The canonical momentum is defined as:

\(\pi = \frac{\partial L}{\partial \dot{q}_i}\)

To obtain Hamilton’s equations, take the total differential of H. You find:

\(dH = \sum_i (\dot{q}_i dp_i + p_i d\dot{q}_i) - \sum_i \left( \frac{\partial L}{\partial q_i} dq_i + \frac{\partial L}{\partial \dot{q}_i} d\dot{q}_i \right)\)

Use the Euler-Lagrange equations, ṗᵢ = ∂L/∂qᵢ, to simplify this. The result is dH = ∑ᵢ (q̇ᵢdpᵢ − ṗᵢdqᵢ). From this, you can read off Hamilton’s equations:

\(\dot{q}_i = \frac{\partial H}{\partial p_i} \ \ \ \ \ \ \ \ \ \\ \ \ \dot{p}_i = -\frac{\partial H}{\partial q_i}\)

Question 4: What are Poisson brackets, and what’s their quantum mechanical analogue?

The Poisson bracket of two functions f and g is:

\(f, g = \sum_i \left( \frac{\partial f}{\partial q_i} \frac{\partial g}{\partial p_i} - \frac{\partial f}{\partial p_i} \frac{\partial g}{\partial q_i} \right)\)

It specifies the time evolution of an observable A. The governing equation is dA/dt = {A, H} + ∂A/∂t. Its quantum analogue is the commutator of two operators. You write this as [Â, B̂] = ÂB̂ − B̂Â. Dirac’s canonical quantization rule relates them. The rule is:

\(\{A, B\} \to \frac{1}{i\hbar}[\hat{A}, \hat{B}]\)

Question 5: Write down Maxwell’s equations in differential form and explain them.

Maxwell’s equations are a set of four differential equations, which describe electromagnetism.

• ∇ ⋅ E = ρ / ε₀ (Gauss’s Law): Electric field flux from a surface equals the enclosed charge.

• ∇ ⋅ B = 0 (Gauss’s Law for Magnetism): It’s stipulated that there are no magnetic monopoles. Magnetic field lines have always been found to make closed loops, so that’s encoded into the laws.

• ∇ × E = −∂B/∂t (Faraday’s Law): A changing magnetic field creates a circulating electric field.

• ∇ × B = μ₀(J + ε₀ ∂E/∂t) (Ampère-Maxwell Law): A magnetic field originates from currents and changing electric fields.

Question 6: Explain gauge invariance in classical EM.

Gauge invariance means the physical fields E and B don’t alter under certain changes. You define fields from potentials: E = −∇φ − ∂A/∂t and B = ∇ × A. A gauge transformation modifies these potentials. The modification looks like:

\(A \rightarrow A + \nabla \chi \ \ \\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \phi \rightarrow \phi - \frac{\partial \chi}{\partial t}\)

Here, χ is any scalar function (usually smooth). The fields E and B stay the same. This shows a redundancy in our description.

Another way of saying something has EM has a gauge symmetry is saying the potentials have more degrees of freedom than are physically required.

Question 7: How are the electric and magnetic fields unified using the electromagnetic field tensor F^μν?

You can unify the electric and magnetic fields into a single object. This is the electromagnetic field tensor F^μν. It’s a part of special relativity’s “covariant” formulation (more on covariance later). The tensor is built from the four-potential A^μ = (φ/c, A). The definition is F^μν = ∂^μA^ν − ∂^νA^μ. In matrix form, its components are:

\(F_{\mu\nu}=\begin{pmatrix} 0 & -E_x & -E_y & -E_z\\ E_x & 0 & -B_z & B_y\\ E_y & B_z & 0 & -B_x\\ E_z & -B_y & B_x & 0 \end{pmatrix}\)

This unification shows that E and B fields aren’t separate. In fact, they change into each other under Lorentz boosts.

Question 8: What’s the ontological interpretation of the wavefunction Ψ(x, t)?

This is a trick question. We don’t have a consensus on the “ontological” interpretation of it, as we don’t have a consensus on interpretations of quantum mechanics in general.

Anyhow, the wavefunction Ψ(x, t) is a function with complex values. It contains the complete quantum state of a particle, but it’s not directly observable itself. Its physical interpretation is probabilistic, given by the Born rule.

This means that the squared modulus, |Ψ(x, t)|², is a probability density, in this case of finding the particle at position x at time t. For consistency, the total probability must be one, leading to the normalization condition:

\(\int_{-\infty}^{\infty} |\Psi(x,t)|^2 \, dx = 1\)

Question 9: Explain the Heisenberg Uncertainty Principle using the commutator of position and momentum operators.

The Uncertainty Principle is a tenet of quantum mechanics. It says you can’t know certain pairs of properties perfectly. For position x̂ and momentum p̂ₓ, their commutator is [x̂, p̂ₓ] = iħ. These are called “complementary variables,” and they don’t need to be position / momentum, generally. The general uncertainty relation is:

\(\sigma_A^2 \sigma_B^2 \geq \left(\frac{1}{2i} \langle [\hat{A}, \hat{B}] \rangle \right)^2\)

For position and momentum, you get the famous result:

\(\sigma_x \sigma_p \geq \frac{\hbar}{2}\)

Thus, if you know position precisely (σₓ → 0), momentum becomes very uncertain (σₚ → ∞). However, I shouldn’t even say “know” since technically speaking, this is a result of the operator algebra and the epistemology is either a consequence of that or the algebra just models our data.

Question 10: Describe the concept of Hilbert space in quantum mechanics.

A Hilbert space ℋ is a complex vector space with an inner product. That’s it. Oh, and it’s also a complete metric space. In quantum mechanics:

• States: Quantum states are represented by vectors, or kets |ψ⟩, in Hilbert space. Density operators are the more general / accurate formalism, but for now, this suffices.

• Observables: Physical quantities are represented by self-adjoint operators. These operators act on the Hilbert space.

• Probabilities: The inner product ⟨φ|ψ⟩ is used to compute probability amplitudes.

The Hilbert space structure is necessary in ordinary quantum mechanics as it allows for these features like superposition, measurement, and operators.

In other formalisms, Hilbert spaces aren’t necessary (though they can be derived). See this podcast with Jacob Barandes.

Question 11: What’s quantum entanglement, and how does it conflict with classical notions of locality?

Entanglement is a quantum link between two or more particles where the whole system has a single wavefunction, and this wavefunction can’t be factored into individual states. Nothing magical. Spooky, maybe, but not magical.

An entangled pair could be in the state (1/√2)(|↑↓⟩ − |↓↑⟩), for instance. If you measure one spin as up, the other is instantly down. It seems to conflict with locality since locality says objects are only influenced by immediate surroundings.

Huge controversies surround Bell’s theorem and its assumptions. See this podcast with Tim Maudlin.

Question 12: Explain the path integral formulation of quantum mechanics.

Feynman’s path integral formulation is another view of quantum mechanics. It says the amplitude for a particle to go from A to B is a sum. This sum is over all possible paths between the points. Each path contributes with a phase factor e^(iS/ħ). Here, S is the classical action for that path. The total amplitude is:

\(K(B, A) = \int D[x(t)] e^{i S[x(t)] / \hbar}\)

This directly connects to classical physics. In the limit ħ → 0, the phase factor oscillates wildly and paths will destructively interfere, except for one. This is the classical path of stationary action, δS = 0.

Note that this doesn’t imply that particles literally take all possible paths. Here’s my detailed explanations.

Question 13: What’s the quantum Zeno effect?

The quantum Zeno effect is such an odd phenomenon. It basically says that a quantum system’s evolution can be frozen by frequent measurements.

Specifically, if a system’s evolving from state |A⟩ to another (different) state |B⟩ and you repeatedly check if the system is still in state |A⟩, you alter its evolution. Each measurement “collapses” the wavefunction back to |A⟩. The transition probability in a small time Δt is proportional to (Δt)².

By making the measurements very frequent, you can actually keep the total transition probability near zero. This freezes the system in its initial state.

Question 14: Derive the time evolution of the expectation value of an operator Â.

This is known as Ehrenfest’s theorem. It relates the time derivative of an expectation value to a commutator. The expectation value is ⟨Â⟩ = ⟨ψ(t)|Â|ψ(t)⟩. You take its time derivative, and then using the product rule, you find:

\(\frac{d\langle \hat{A} \rangle}{dt} = \langle \frac{\partial \psi}{\partial t} | \hat{A} | \psi \rangle + \langle \psi | \frac{\partial \hat{A}}{\partial t} | \psi \rangle + \langle \psi | \hat{A} | \frac{\partial \psi}{\partial t} \rangle\)

Now, use the Schrödinger equation iħ ∂|ψ⟩/∂t = Ĥ|ψ⟩ and its conjugate. This gives d⟨Â⟩/dt = (1/iħ)⟨ψ|(ÂĤ − ĤÂ)|ψ⟩ + ⟨∂Â/∂t⟩.

The final result is:

\(\frac{d\langle \hat{A} \rangle}{dt} = \frac{1}{i\hbar} \langle [\hat{A}, \hat{H}] \rangle + \langle \frac{\partial \hat{A}}{\partial t} \rangle\)

Question 15: Define the canonical partition function Z and explain how thermodynamic quantities can be derived from it.

The canonical partition function Z describes a system at temperature T. It’s the summation over all possible microstates s, where each state is weighted by its Boltzmann factor, e^(−βEₛ). Here, β = 1/(kᵦT) and Eₛ is the state’s energy. This, by the way, is the connection between imaginary time and inverse temperature that you hear about, but more on that later.

\(Z \equiv \sum_s e^{-\beta E_s}\)

The partition function is a “generating function” for thermodynamic quantities, meaning that you can get other quantities by some closed form formula involving Z.

Helmholtz Free Energy:

\(F = -k_B T \ln Z\)

Internal Energy:

\(U = -\frac{\partial \ln Z}{\partial \beta}\)

Entropy:

\(S = k_B (\ln Z + \beta U)\)

These are just some examples.

Question 16: What’s the distinction between the microcanonical, canonical, and grand canonical ensembles?

This confused me for a while, so let’s delineate:

• Microcanonical Ensemble: This depicts an isolated system, with fixed energy E, volume V, and particle number N. All allowed microstates are considered equally probable. It’s foundational but often difficult to work with for various reasons.

• Canonical Ensemble: Depicts a system in thermal contact with a heat bath. It has fixed N, V, and temperature T. Energy can be exchanged. The probability of a state is proportional to e^(−Eₛ/kᵦT). This one tends to be worked with more frequently in theoretical physics (e.g., black hole thermodynamics, coming up later down the page).

• Grand Canonical Ensemble: This guy depicts an open system. It can exchange both energy and particles with a reservoir. It has fixed V, T, and chemical potential μ.

Question 17: Explain spontaneous symmetry breaking using the Ising model as an example.

Spontaneous symmetry breaking is what happens when a system’s ground state is less symmetric than the fundamental laws that gave rise to it. The 2D Ising model is a specific simple illustration, usually encountered first. Its Hamiltonian is symmetric under a global spin-flip (sᵢ → −sᵢ):

\(H \;=\; -J \sum_{\langle i,j\rangle} s_i\,s_j \)

• Above the critical temperature (T > T_c): The system is symmetric. The average magnetization is zero, ⟨M⟩ = 0.

• Below the critical temperature (T < T_c): The system has to choose a ground state. It picks either all spins up or all spins down. The resulting state isn’t symmetric, even though the Hamiltonian is! Thus the symmetry is (“spontaneously”) broken.

Question 18: What’s entropy from a statistical mechanics point of view, according to Boltzmann?

From a statistical mechanics standpoint, entropy S measures something simple. It’s the number of microscopic states, Ω, that look the same macroscopically. Boltzmann’s famous formula connects them:

\(S = k_B \ln \Omega\)

Here, kᵦ is the Boltzmann constant. A state with higher entropy can be realized in more ways than one with a lower entropy, just by definition. This gives a microscopic basis for the Second Law of Thermodynamics. Isolated systems evolve to the macrostate with the largest Ω because that macrostate is overwhelmingly the most probable. That’s why it’s a statistical law and not an inevitability.

Question 19: State the equivalence principle and its role in the formulation of general relativity.

The equivalence principle (precisely, the Einstein equivalence principle; the strong version additionally requires WEP to hold for self-gravitating bodies and for local gravitational experiments) has three parts.

1. Weak Equivalence Principle: A body’s path in a gravitational field is independent of its composition. This means inertial mass equals gravitational mass. This was what even Newton believed (technically m_g ∝ m_i; strict numerical equality only holds after a unit choice).

2. Local Lorentz Invariance: All non-gravitational physics reduce to special relativity in every freely falling frame. Newton didn’t know about this.

3. Local Position Invariance: The outcome of every local non-gravitational experiment is independent of location (and time) in spacetime. The fact that G is treated as a universal constant in Newtonian gravity suggests he would not have objected to this.

The principle’s main role is to equate gravity with acceleration locally. This is what leads one to depict gravity as spacetime curvature (or, in equivalent reformulations, as torsion, non-metricity, etc.) rather than as a Newtonian “force.”

Keep in mind, you can generalize what you mean by “force” and have that gravity is force. See this interview with Claudia de Rham.

Question 20: Write down the Einstein Field Equations and explain the physical meaning of each term.

The Einstein Field Equations show how mass-energy curves spacetime. They are written as:

\(R_{\mu\nu} - \frac{1}{2}R g_{\mu\nu} + \Lambda g_{\mu\nu} = \frac{8\pi G}{c^4} T_{\mu\nu}\)

• R_μν is the Ricci curvature tensor (how volumes change).

• R is the Ricci scalar. It’s the overall curvature represented as one real number (at a point).

• g_μν is the metric tensor (defines distances in spacetime).

• Λ is the cosmological constant (in some cosmological models, it represents vacuum energy density).

• T_μν is the stress-energy tensor. It’s the source of gravity (because of nonlinearity, the metric itself is another source though).

The left side is geometry. The right side is matter / energy.

Note, people will say the metric defines distances and angles, but the angles are calculated from the distances, so infinitesimal distance is more fundamental.

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• Question 21: What’s a black hole singularity? What does the Cosmic Censorship Hypothesis propose about it specifically?

• Question 22: Describe what a gravitational wave is and how its existence is predicted by GR.

• Question 23: What’s a conformal field theory?

• Question 24: What’s the Penrose process for extracting energy from a rotating black hole?

• Question 25: What’s the difference between the Klein-Gordon and Dirac equations?

• Question 26: How are particles interpreted in QFT?

• Question 27: What do Feynman diagrams represent?

• Question 28: Explain renormalization in QFT and why it’s necessary.

• Question 29: What’s a gauge theory? Provide QED as an example.

• Question 30: Explain the concept of asymptotic freedom in QCD.

• Question 31: What’s the Higgs mechanism and what problem does it solve in the Standard Model?

• Question 32: What are Goldstone bosons and how do they relate to spontaneous symmetry breaking?

• Question 33: What role does group theory, specifically Lie groups like SU(2) and SU(3), play in particle physics?

• Question 34: What’s a fiber bundle, and how is it relevant to gauge theories?

• Question 35: What’s the primary motivation for string theory?

• Question 36: What’s the Atiyah-Singer index theorem, and what’s its conceptual significance?

• Question 37: Explain the connection between the path integral in quantum mechanics and the Wiener measure in stochastic processes.

• Question 38: What’s supersymmetry?

• Question 39: How do you calculate the dimensions that the string theories live in?

• Question 40: Explain the holographic principle and its realization in the AdS / CFT correspondence.

• Question 41: What are the leading candidates for dark matter?

• Question 42: What’s dark energy, and what’s its proposed role in the expansion of the universe?

• Question 43: What are the primary (conceptual) difficulties in formulating a theory of quantum gravity?

• Question 44: What’s the black hole information paradox?

• Question 45: Describe the concept of a spin foam in loop quantum gravity.

• Question 46: What’s M-theory and how does it “unify” (relate) the five superstring theories?

• Question 47: Prove that a massless spin 2 field (Lorentz-invariant, interacting) gives rise to general relativity in the classical limit.

• Question 48: What defines a topological quantum field theory? And give two examples.

• Question 49: How are fermions unified in SO(10)?

• Question 50: Explain the CPT theorem and its implications for the relationship between matter and antimatter.

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—Curt Jaimungal

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