Path Integrals

Introduction

The previous lectures pushed Hamiltonian perturbation theory in QED far enough to reveal both its power and its limitations.

On the one hand, the Hamiltonian formalism gave a fully defined quantum theory:

• photons, electrons, and positrons are operator excitations in one Hilbert space,
• the interaction Hamiltonian is explicit,
• and in principle every process can be computed by ordinary quantum-mechanical perturbation theory.

On the other hand, even very simple-looking processes become algebraically ugly. The lecture reviews Compton scattering and reminds us that at leading order in $\alpha$, the amplitude comes from four separate time-ordered operator terms. Even worse, those four “processes” are not Lorentz invariant separately; only after they are added in pairs do Lorentz-invariant amplitudes appear. :contentReference[oaicite:1]{index=1}

That is the central motivation for this lecture.

The problem is not only divergence. The problem is also that the Hamiltonian formulation inevitably hides both Lorentz invariance and gauge invariance. Those symmetries are still present in the theory, but they are buried inside long operator calculations and mysterious cancellations. If we want a clearer and more efficient perturbation theory, we need a formulation in which:

• the basic object is the Lagrangian, which is naturally a Lorentz scalar,
• the gauge degrees of freedom can remain visible,
• and perturbation theory can be generated without hand-managing endless operator reorderings.

That is exactly what the path-integral formalism does. :contentReference[oaicite:2]{index=2}

This lecture does not jump straight into QED path integrals. Instead it builds the formalism from the ground up in progressively more useful cases:

1. the original Feynman–Hibbs path integral for one canonical degree of freedom using $q$- and $p$-eigenstates,
2. the coherent-state path integral for a bosonic harmonic oscillator,
3. the fermionic coherent-state path integral using Grassmann numbers.

The lecture is very honest that for a general nonlinear quantum-mechanical Hamiltonian, path integrals are usually not easier than operator methods. But for quadratic systems with linear sources, they are powerful, and more importantly they provide generating-function machinery that builds all of perturbation theory. That is exactly what field theory needs. :contentReference[oaicite:3]{index=3}

So the point of this lecture is not “path integrals are mystical sums over all histories.” The point is much more concrete: path integrals are an equivalent Lagrangian reformulation of quantum mechanics that prepares a cleaner perturbation theory for relativistic quantum fields.

Learning Objectives

• Explain why the complexity of Hamiltonian perturbation theory motivates a Lagrangian formulation.
• Derive the Feynman–Hibbs phase-space path integral from repeated insertion of the identity operator.
• Understand the path integral as the continuum notation for a limit of many finite-dimensional integrals.
• Recognize the quantity

$$\int dt\,[p\dot q - H(q,p)]$$

as the Hamiltonian-action form of the Lagrangian principle.

• Define unnormalized bosonic coherent states and their resolution of the identity.
• Construct the coherent-state path integral for a bosonic oscillator.
• Understand why path integrals are especially useful for quadratic Lagrangians with linear sources.
• Solve the driven harmonic oscillator in coherent-state path-integral form.
• Explain how source differentiation generates perturbation theory.
• Define Grassmann numbers and explain why they are needed for fermionic path integrals.
• Construct the fermionic coherent-state identity resolution.
• Write the fermionic coherent-state path integral for a two-state system.
• Explain why bosonic and fermionic path integrals are structurally parallel.
• Understand why this formalism is the correct preparation for Lagrangian perturbation theory in QFT.

Prerequisite Knowledge

• Hamiltonian and Lagrangian formulations of mechanics
• Time evolution operator and time ordering
• Harmonic oscillator creation and annihilation operators
• Coherent states
• Basic perturbation theory
• Dirac notation
• Basic calculus of complex variables
• Canonical anticommutation relations for fermions

1. Motivation: why leave the Hamiltonian formalism?

The lecture begins by reviewing exactly what went wrong in the Hamiltonian approach.

Compton scattering at leading order in $\alpha$ required four separate time-ordered operator contributions, each corresponding to a different intermediate-state process with different particle content. These process amplitudes were not Lorentz invariant separately; Lorentz invariance only emerged after the right sums were taken. The same review slide also reminds us that stationary perturbation theory for vacuum energy and one-particle energies leads to divergent integrals whose organization is not at all transparent. :contentReference[oaicite:4]{index=4}

The lecture then states the motivation clearly:

• the divergences are real,
• but the huge algebraic complexity and the mysterious conspiracies of cancellations come from the Hamiltonian formulation,
• because the Hamiltonian formulation inevitably hides Lorentz invariance and gauge invariance.

So the proposed cure is: construct a new Lagrangian formulation of quantum mechanics that is equivalent to the Hamiltonian one, but based on Lagrangians rather than Hamiltonians. Since Lagrangians in relativistic field theory are Lorentz scalars, this gives a more symmetry-friendly language for perturbation theory. :contentReference[oaicite:5]{index=5}

That is the conceptual starting point.

2. Lecture plan

The lecture outline is simple and exact:

1. Feynman–Hibbs path integral
2. Coherent-state path integral
3. Fermionic path integral :contentReference[oaicite:6]{index=6}

This ordering matters. The lecture wants to show that the bosonic and fermionic path integrals are not foreign inventions. They are direct analogues of the same repeated-identity construction.

3. Feynman–Hibbs path integrals: one canonical degree of freedom

The lecture begins with the original phase-space path integral for one ordinary quantum-mechanical degree of freedom, with operators $\hat q$ and $\hat p$, their eigenstates $|q\rangle$ and $|p\rangle$, and the identity resolution

$$\frac{1}{2\pi}\int dq\,dp\; e^{iqp/\hbar} |q\rangle\langle p| = \hat 1.$$

Using this identity, the lecture states the path-integral representation

$$\langle p_f|\hat U(t_f,t_i)|q_i\rangle = \int \mathcal Dq\,\mathcal Dp\; \exp\left[ \frac{i}{\hbar}\int_{t_i}^{t_f} dt\, \big(p\dot q - H(q,p)\big) \right].$$

It then immediately asks: what does this really mean? :contentReference[oaicite:7]{index=7}

That question is important because the lecture does not want the student to treat the path integral as mystical notation. It wants to derive it as a limit of ordinary integrals.

4. Time evolution as many short steps

The lecture rewrites the time-evolution operator as a time-ordered product of many short exponential factors:

$$\hat U(t_f,t_i) = \lim_{N\to\infty} e^{- \frac{i}{\hbar}\frac{T}{N}\hat H(t_f)} e^{- \frac{i}{\hbar}\frac{T}{N}\hat H(t_f-\frac{T}{N})} \cdots e^{- \frac{i}{\hbar}\frac{T}{N}\hat H(t_i+\frac{T}{N})}, \qquad T=t_f-t_i.$$

Between every adjacent pair of short-time factors, it inserts the identity operator in the mixed $q,p$ form above. That is the main construction idea: path integrals come from repeated insertion of complete sets of states between many short time-evolution steps. :contentReference[oaicite:8]{index=8}

This is the first really important demystification.

5. Small-time matrix elements and normal ordering in $\hat p$

The lecture then chooses a convenient operator ordering convention:

$$\hat H(t) = :H(\hat q,\hat p,t):$$

meaning that all $\hat p$ operators are written to the left. It notes that this can always be done for the right $c$-number function $H$.

For one short step, expanding to first order in $T/N$ gives

$$\langle p_n| e^{- \frac{i}{\hbar}\frac{T}{N}\hat H(t_n)} |q_{n-1}\rangle = e^{- \frac{i}{\hbar}\frac{T}{N} H(q_{n-1},p_n,t_n)}\, \langle p_n|q_{n-1}\rangle + O((T/N)^2).$$

Using

$$\langle p|q\rangle = \frac{1}{\sqrt{2\pi\hbar}}e^{-iqp/\hbar},$$

the product of all short-step factors becomes one large finite-dimensional integral over the intermediate $q_n,p_n$. :contentReference[oaicite:9]{index=9}

This is how the exponential of the action appears.

6. The path integral is just shorthand for a limit

After combining the phases, the lecture obtains

$$\langle p_f|\hat U(t_f,t_i)|q_i\rangle = \lim_{N\to\infty} \int \frac{d^{N-1}q\,d^{N-1}p}{(2\pi\hbar)^N} \exp\left[ \frac{i}{\hbar} \sum_{n=1}^N \left( (q_n-q_{n-1})p_n - \frac{T}{N}H(q_{n-1},p_n,t_n) \right) \right].$$

Then it introduces interpolating functions $q(t)$ and $p(t)$, observes that the sum is a Riemann sum for

$$\int_{t_i}^{t_f} dt\, [p\dot q - H(q,p)],$$

and writes the formal continuum notation

$$\langle p_f|\hat U(t_f,t_i)|q_i\rangle = \int \mathcal Dq\,\mathcal Dp\; e^{\frac{i}{\hbar}\int dt\, [p\dot q-H(q,p)]}.$$

But it immediately clarifies: the so-called path integral is simply notation for the finite-dimensional limit above. :contentReference[oaicite:10]{index=10}

This is one of the cleanest statements in the lecture: a path integral is not magic. It is a compact label for a limiting procedure.

7. Why the lecture does not dwell on the original $q$-$p$ form

The lecture then says that instead of exploring the original Feynman–Hibbs form further, it will move to a different but equivalent path integral that is more useful for field theory: the coherent-state path integral. :contentReference[oaicite:11]{index=11}

This is because bosonic quantum fields are naturally built from harmonic oscillators, and coherent states are the oscillator-adapted basis.

8. Bosonic coherent states

The lecture defines the unnormalized coherent states of an oscillator:

$$|\alpha\rangle = \sum_{n=0}^\infty \frac{\alpha^n}{n!}|n\rangle,$$

so that

$$\hat a |\alpha\rangle = \alpha |\alpha\rangle.$$

Their overlap is

$$\langle \bar\beta|\alpha\rangle = e^{\bar\beta \alpha}.$$

The resolution of the identity is

$$\int \frac{d^2\alpha}{2\pi}\, e^{- \frac12 \bar\alpha \alpha} |\alpha\rangle \langle \bar\alpha| = \hat 1.$$

The lecture writes $d^2\alpha$ both in Cartesian and polar form and emphasizes that $\alpha$ is an ordinary complex number here. :contentReference[oaicite:12]{index=12}

This is the coherent-state analogue of the earlier $|q\rangle,|p\rangle$ identity resolution.

9. Bosonic coherent-state path integral

Now take a Hamiltonian written in normal-ordered oscillator form:

$$\hat H(t)=:H(\hat a,\hat a^\dagger,t):.$$

Inserting coherent-state identities between many short-time evolution factors gives the coherent-state path integral

$$\langle \bar\alpha_f|\hat U(t_f,t_i)|\alpha_i\rangle = \int \mathcal D\alpha\,\mathcal D\bar\alpha\; e^{\frac12(\bar\alpha_f\alpha(t_f)+\bar\alpha(t_i)\alpha_i)} \exp\left[ \int_{t_i}^{t_f} dt \left( \frac{\dot{\bar\alpha}\alpha - \bar\alpha\dot\alpha}{2} -\frac{i}{\hbar}H(\alpha,\bar\alpha,t) \right) \right].$$

The lecture is very candid here: for a general nonlinear Hamiltonian, evaluating this path integral is usually not easier than solving the Schrödinger equation by ordinary operator methods. In fact it is often harder. :contentReference[oaicite:13]{index=13}

That honesty matters. The point is not that path integrals are universally easier; it is that they organize perturbation theory differently.

10. Driven harmonic oscillator as the key solvable case

The lecture then chooses the case that really matters for future field theory: the driven harmonic oscillator,

$$\hat H(t)=\hbar\omega \hat a^\dagger \hat a + \hbar J(t)\hat a^\dagger + \hbar \bar J(t)\hat a.$$

For this Hamiltonian, the coherent-state path integral is exactly solvable, and the lecture gives the result:

$$\langle \bar\alpha_f|\hat U(t_f,t_i)|\alpha_i\rangle = e^{\bar\alpha_f \alpha_i e^{-i\omega(t_f-t_i)}} e^{-i\int dt\,[\alpha_i \bar J(t)e^{-i\omega(t-t_i)}+\bar\alpha_f J(t)e^{-i\omega(t_f-t)}]} e^{-\int_{t_i}^{t_f}dt\int_{t_i}^{t}dt'\,\bar J(t)J(t')e^{-i\omega(t-t')}}.$$

The lecture jokingly says the calculation is “easy,” then spends nine yellow pages doing it. :contentReference[oaicite:14]{index=14}

This is not a digression. This exact generating function is what later makes perturbation theory efficient.

11. Shifting the integration variables

To solve the path integral, the lecture shifts the variables:

$$\alpha(t)=\alpha_0(t)+\Delta(t), \qquad \bar\alpha(t)=\bar\alpha_0(t)+\bar\Delta(t).$$

The idea is to choose $\alpha_0,\bar\alpha_0$ so that all terms linear in $\Delta,\bar\Delta$ vanish. The boundary conditions are chosen as

$$\alpha_0(t_i)=\alpha_i, \qquad \bar\alpha_0(t_f)=\bar\alpha_f, \qquad \Delta(t_i)=0, \qquad \bar\Delta(t_f)=0.$$

The lecture also points out a subtle but important fact: because the integrand is analytic, $\alpha$ and $\bar\alpha$ need not be treated as literal complex conjugates during the deformation. They are independent integration variables in the path-integral sense. :contentReference[oaicite:15]{index=15}

This is a crucial technical habit for later field theory.

12. Classical-looking equations from the path integral

After the shift, the lecture chooses $\alpha_0,\bar\alpha_0$ to satisfy

$$\dot{\bar\alpha}_0 - i\omega \bar\alpha_0 = i\bar J, \qquad \dot\alpha_0 + i\omega \alpha_0 = -iJ.$$

These are exactly the simple driven equations one would write for classical oscillator amplitudes. With this choice, the linear terms disappear and the exponent simplifies drastically. :contentReference[oaicite:16]{index=16}

This is the deeper lesson: path integrals often reduce the quantum calculation to solving classical-looking equations of motion with source terms.

13. The remaining fluctuation path integral equals 1

After substituting the solutions for $\alpha_0,\bar\alpha_0$, the path integral factorizes into:

• a source- and boundary-dependent exponential,
• times a remaining path integral over $\Delta,\bar\Delta$.

The lecture then evaluates this leftover factor indirectly by considering the special case

$$\alpha_i=\bar\alpha_f=J=\bar J=0.$$

Then the Hamiltonian is simply $\hbar\omega \hat a^\dagger \hat a$, and the transition amplitude must be

$$\langle 0|e^{-i\omega(t_f-t_i)\hat a^\dagger \hat a}|0\rangle = 1.$$

Therefore the remaining fluctuation path integral is exactly 1. :contentReference[oaicite:17]{index=17}

This is a very elegant trick: the fluctuation determinant is fixed by normalization rather than by direct integration.

14. Cross-check by differential equation

The lecture then cross-checks the result by differentiating with respect to $t_f$. It shows that the derived expression obeys the correct first-order Schrödinger evolution equation and the correct initial condition

$$\langle \bar\alpha_f|\hat U(t_i,t_i)|\alpha_i\rangle = e^{\bar\alpha_f \alpha_i}.$$

Therefore the coherent-state path-integral result is exactly correct. :contentReference[oaicite:18]{index=18}

So the lecture has now solved an arbitrarily driven oscillator in a path-integral language.

15. Why this helps with QFT

The lecture then states explicitly why this calculation matters for field theory.

The exponent in the coherent-state path integral is the Lagrangian. More importantly, even though exact path-integration is only easy for quadratic Lagrangians with linear time-dependent driving, that is already enough to generate perturbation theory:

$$e^{\frac{i}{\hbar}\int dt\, (L_0+\epsilon L_1)} = \sum_{n=0}^\infty \left(\frac{i\epsilon}{\hbar}\right)^n \left[ \int dt\, L_1\!\left(\frac{\delta}{\delta \bar J(t)},\frac{\delta}{\delta J(t)}\right) \right]^n e^{\frac{i}{\hbar}\int dt\,(L_0 + J\bar\alpha + \bar J\alpha)} \Bigg|_{J=0}.$$

So once we know the generating functional for the quadratic driven problem, we can build perturbation theory for more complicated interactions by functional differentiation with respect to the sources. :contentReference[oaicite:19]{index=19}

This is the main practical reason path integrals become so powerful in QFT.

16. Why fermions need something new

The lecture then turns to fermionic fields.

Bosonic quantum fields are large collections of harmonic oscillators, so bosonic coherent states fit them naturally. But fermionic fields are built from many two-state systems obeying anticommutation relations, not ordinary oscillator commutators. To construct a parallel path integral, the lecture says, we need “coherent states” whose eigenvalues are Grassmann numbers.

Grassmann numbers are not operators in Hilbert space, but they are not ordinary $c$-numbers either. They anticommute:

$$\eta_1\eta_2 = -\eta_2\eta_1, \qquad \eta^2 = 0.$$

That nilpotency is the key algebraic feature. :contentReference[oaicite:20]{index=20}

So fermionic path integrals require a new kind of external algebra.

17. Fermionic coherent states for a two-state system

Consider a two-state Hilbert space $|0\rangle,|1\rangle$ with lowering and raising operators

$$\hat b = |0\rangle\langle 1|, \qquad \hat b^\dagger = |1\rangle\langle 0|.$$

The lecture defines the fermionic coherent state

$$|\eta\rangle = \sum_{n=0}^1 \eta^n |n\rangle = |0\rangle + \eta |1\rangle.$$

Then

$$\hat b|\eta\rangle = \eta |\eta\rangle.$$

So the fermionic annihilation operator has a Grassmann eigenvalue. :contentReference[oaicite:21]{index=21}

This is the direct fermionic analogue of the bosonic coherent-state definition.

18. Grassmann calculus

The lecture then introduces a minimal Grassmann calculus. Since $\eta^2=\bar\eta^2=0$, any function of two Grassmann variables has only four terms:

$$F(\eta,\bar\eta)=A + B\bar\eta + \bar B \eta + C \eta \bar\eta.$$

Differentiation is defined algebraically, e.g.

$$\frac{\partial F}{\partial \eta} = \bar B + C\bar\eta, \qquad \frac{\partial F}{\partial \bar\eta} = B - C\eta.$$

Then the lecture defines Grassmann integration by the strange-looking but very useful rule

$$\int d\eta\, F = \frac{\partial F}{\partial \eta}, \qquad \int d\bar\eta\, F = \frac{\partial F}{\partial \bar\eta}, \qquad \int d\bar\eta\, d\eta\, F = \frac{\partial^2 F}{\partial \bar\eta\, \partial \eta}.$$

The lecture is explicit: the only reason for this definition is that it gives the right identity operator. :contentReference[oaicite:22]{index=22}

This is an important philosophical point. Grassmann integration is defined by utility, not by geometric intuition.

19. Fermionic identity resolution

Using that Grassmann calculus, the lecture shows

$$\int d\bar\eta\, d\eta\; |\eta\rangle\langle \bar\eta|\, e^{-\bar\eta\eta} = |0\rangle\langle 0| + |1\rangle\langle 1| = \hat 1.$$

This is the exact fermionic analogue of the bosonic coherent-state identity resolution. :contentReference[oaicite:23]{index=23}

So now the same repeated-identity construction can be repeated for fermionic time evolution.

20. Fermionic coherent-state path integral

The lecture then writes the fermionic coherent-state path integral for a two-state system:

$$\langle \bar\eta_f|\hat U(t_f,t_i)|\eta_i\rangle = \int \mathcal D\eta\,\mathcal D\bar\eta\; e^{\frac12(\bar\eta_f \eta(t_f)+\bar\eta(t_i)\eta_i)} \exp\left[ \frac{i}{\hbar}\int_{t_i}^{t_f} dt \left( \hbar\frac{\dot{\bar\eta}\eta - \bar\eta\dot\eta}{2i} - H(\eta,\bar\eta,t) \right) \right].$$

The lecture comments that this comes out exactly the same in form as the bosonic coherent-state path integral, by construction. :contentReference[oaicite:24]{index=24}

That structural parallel is one of the main takeaways:

• bosons: complex commuting variables,
• fermions: Grassmann anticommuting variables,
• same coherent-state logic.

21. The driven fermionic two-state system

The lecture then takes the fermionic analogue of the driven oscillator,

$$\hat H(t) = \hbar\omega \hat b^\dagger \hat b + \bar J(t)\hat b + J(t)\hat b^\dagger,$$

where now $J,\bar J$ are Grassmann-valued sources. It states the exact transition amplitude:

$$\langle \bar\eta_f|\hat U(t_f,t_i)|\eta_i\rangle = e^{\bar\eta_f \eta_i e^{-i\omega(t_f-t_i)}} e^{- \frac{i}{\hbar}\int dt\,[\bar\eta_f e^{-i\omega(t_f-t)}J(t) + \bar J(t)\eta_i e^{-i\omega(t-t_i)}]} e^{- \frac{1}{\hbar^2}\int dt \int_{t_i}^{t} dt' \,\bar J(t)J(t')e^{-i\omega(t-t')}}.$$

The lecture says this can be shown either by the path integral or by ordinary methods. :contentReference[oaicite:25]{index=25}

This is the fermionic generating functional analogue of the driven bosonic result.

22. Why this completes the preparation for QFT

The lecture ends by pointing out the exact analogy:

For bosons, the coherent-state path integral plus source differentiation generates perturbation theory.

For fermions, the Grassmann coherent-state path integral does the same, provided the sources are Grassmann-valued.

So now we have both ingredients needed for quantum field theory:

• bosonic path integrals for oscillator-like fields,
• fermionic path integrals for anticommuting fields,
• and both in a Lagrangian formulation that is much better suited to Lorentz-covariant perturbation theory than the canonical Hamiltonian approach. :contentReference[oaicite:26]{index=26}

That is the whole point of the lecture.

Worked Examples

Example 1: Why the path integral is really a limit, not a mystical object

Starting from

$$\hat U(t_f,t_i) = \lim_{N\to\infty}\prod_{n=1}^N e^{- \frac{i}{\hbar}\frac{T}{N}\hat H(t_n)},$$

insert the identity

$$\frac{1}{2\pi}\int dq\,dp\, e^{iqp/\hbar}|q\rangle\langle p| = \hat 1$$

between every pair of short-time factors. This produces an ordinary finite-dimensional integral over all intermediate $q_n,p_n$. Only after writing the resulting exponent as a Riemann sum do we introduce the shorthand

$$\int \mathcal Dq\,\mathcal Dp\; e^{\frac{i}{\hbar}\int dt\,[p\dot q-H(q,p)]}.$$

So the “path integral” is just the notation for the $N\to\infty$ limit of ordinary integrals.

Example 2: Why Grassmann integration is defined the way it is

For a fermionic coherent state $|\eta\rangle = |0\rangle + \eta |1\rangle$, we want an identity operator of the form

$$\int d\bar\eta\, d\eta\; |\eta\rangle\langle \bar\eta|\, e^{-\bar\eta\eta}.$$

This works only if Grassmann integration is defined so that

$$\int d\eta\, 1 = 0, \qquad \int d\eta\, \eta = 1,$$

and similarly for $\bar\eta$. With that definition, the integral reproduces exactly

$$|0\rangle\langle 0| + |1\rangle\langle 1| = \hat 1.$$

So Grassmann integration is designed precisely to make the coherent-state identity resolution work.

Intuition

The intuition of the lecture is simple but powerful:

Hamiltonian perturbation theory keeps forcing us to manipulate operators directly, and that hides the symmetries we care most about in relativistic field theory. Path integrals reorganize the same quantum mechanics in terms of the action, which is built from the Lagrangian and therefore carries Lorentz structure much more naturally.

For bosons, the natural variables are coherent-state amplitudes. For fermions, the same idea works only if we allow anticommuting Grassmann numbers. In both cases, the exact solvable object is the quadratic theory with sources. Once that is solved, perturbation theory for more complicated interactions comes from taking source derivatives.

So the path integral is not replacing quantum mechanics. It is repackaging it into a form that is much better adapted to field theory.

Common Mistakes

• Thinking the path integral is a fundamentally different theory from ordinary quantum mechanics. The lecture insists it is an equivalent reformulation.
• Treating the path integral symbol $\int \mathcal Dq\,\mathcal Dp$ as more fundamental than the finite-$N$ limit it abbreviates.
• Assuming path integrals are always easier than operator methods. The lecture explicitly says this is false for generic nonlinear systems.
• Forgetting that coherent-state path integrals are the form most useful for bosonic field theory, not the original $q$-$p$ form.
• Thinking $\alpha$ and $\bar\alpha$ must remain literal complex conjugates under every path-integral manipulation. In the analytic continuation/deformation sense they are treated independently.
• Missing that the remaining fluctuation path integral in the driven oscillator is fixed by normalization rather than direct evaluation.
• Treating Grassmann numbers like ordinary commuting numbers. They anticommute and square to zero.
• Thinking Grassmann integration has an intuitive geometric meaning like ordinary integration. Here it is defined algebraically to make the formalism work.
• Forgetting that the fermionic path integral is structurally parallel to the bosonic one.

Short Summary

The lecture begins by reviewing the complexity of Hamiltonian perturbation theory in QED: even Compton scattering at leading order requires four time-ordered terms whose individual amplitudes are not Lorentz invariant, and the divergent stationary perturbative corrections to vacuum and particle energies are organized very awkwardly in the Hamiltonian formalism. It then motivates a new but equivalent formulation of quantum mechanics based on Lagrangians rather than Hamiltonians, precisely because Lagrangians are Lorentz scalars and retain the gauge structure that had to be hidden in the canonical QED Hamiltonian. The lecture first derives the original Feynman–Hibbs phase-space path integral for a single degree of freedom by slicing the time-evolution operator into many short steps and inserting the mixed $q,p$ identity operator between them, obtaining

$$\langle p_f|\hat U(t_f,t_i)|q_i\rangle = \int \mathcal Dq\,\mathcal Dp\; e^{\frac{i}{\hbar}\int dt\,[p\dot q - H(q,p)]},$$

with the important clarification that this path integral is simply shorthand for the $N\to\infty$ limit of ordinary finite-dimensional integrals. The lecture then replaces the $q,p$ representation by bosonic coherent states, deriving the coherent-state path integral

$$\langle \bar\alpha_f|\hat U(t_f,t_i)|\alpha_i\rangle = \int \mathcal D\alpha\,\mathcal D\bar\alpha\; e^{\frac12(\bar\alpha_f\alpha(t_f)+\bar\alpha(t_i)\alpha_i)} e^{\int dt[(\dot{\bar\alpha}\alpha-\bar\alpha\dot\alpha)/2 - \frac{i}{\hbar}H(\alpha,\bar\alpha,t)]}.$$

It solves explicitly the driven harmonic oscillator with source terms, showing that path integrals are especially useful for quadratic systems and that source differentiation then generates perturbation theory. Finally, the lecture constructs the fermionic analogue by introducing Grassmann numbers, fermionic coherent states, and Grassmann integration, which produce the fermionic coherent-state identity operator and the fermionic path integral

$$\langle \bar\eta_f|\hat U(t_f,t_i)|\eta_i\rangle = \int \mathcal D\eta\,\mathcal D\bar\eta\; e^{\frac12(\bar\eta_f\eta(t_f)+\bar\eta(t_i)\eta_i)} e^{\frac{i}{\hbar}\int dt[\hbar(\dot{\bar\eta}\eta-\bar\eta\dot\eta)/(2i)-H(\eta,\bar\eta,t)]}.$$

The lecture closes by emphasizing that this bosonic-plus-fermionic path-integral machinery is exactly what is needed to build a Lagrangian perturbation theory for QFT. :contentReference[oaicite:27]{index=27}

Practice Problems

1. Why does the lecture say that the Hamiltonian formulation hides Lorentz and gauge invariance?

2. In what precise sense is the path integral equivalent to ordinary quantum mechanics rather than a different theory?

3. Why does the lecture insist that the phase-space path integral is really just shorthand for a limit of many ordinary integrals?

4. What is the role of the factor

$$\int dt\,[p\dot q - H(q,p)]$$

in the Feynman–Hibbs construction?

5. Why are coherent states a more useful basis for bosonic field theory than the original $q,p$ basis?

6. Why are path integrals especially useful for quadratic systems with linear sources?

7. What is the purpose of shifting

$$\alpha = \alpha_0 + \Delta,\qquad \bar\alpha = \bar\alpha_0 + \bar\Delta$$

in the driven harmonic-oscillator path integral?

8. Why can the remaining fluctuation path integral be identified with 1 in the bosonic calculation?

9. What are Grassmann numbers, and why are they necessary for fermionic path integrals?

10. Why is Grassmann integration defined in the apparently strange way shown in the lecture?

11. How does the fermionic coherent-state identity operator parallel the bosonic one?

12. Why does solving the quadratic-source problem give all of perturbation theory once source differentiation is introduced?