2  Path Integral of Lattice Discretized Scalar Field Theory

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2.1 Transfer Matrix and Trotter Formula

The Hamiltonian on the 3d lattice splits as \(\hat{H} = \hat{K} + \hat{V}\), where \[ \hat{K} = \sum_{\mathbf{x}} \frac{a^3}{2}\hat{\pi}^2(\mathbf{x}), \qquad \hat{V} = V(\hat{\varphi}) = \sum_{\mathbf{x}} a^3\!\left[\frac{1}{2}(\partial_k^f\hat{\varphi})^2 + \frac{m^2}{2}\hat{\varphi}^2 + \frac{1}{4!}\hat{\varphi}^4\right]\!(\mathbf{x}). \]

The key identity connecting the Euclidean evolution operator to repeated single-step factors is the Trotter formula (strong limit): \[ e^{-T\hat{H}} = \left(e^{-\frac{T}{M}\hat{H}}\right)^{\!M} = \lim_{M\to\infty} \left(e^{-\frac{T}{M}\hat{K}}\,e^{-\frac{T}{M}\hat{V}}\right)^{\!M}. \]

Setting \(T\) = total Euclidean time and \(\tau = T/M\) = time step (so \(\tau \to 0\) as \(M \to \infty\)), the small-\(\tau\) expansion gives \[ e^{-\tau\hat{H}} \approx e^{-\tau(\hat{K}+\hat{V})} \xrightarrow{\tau\to 0} 1 - \tau(\hat{K}+\hat{V}) + O(\tau^2) = \{1-\tau\hat{K}\}\{1-\tau\hat{V}\} + O(\tau^2) = e^{-\tau\hat{K}}\,e^{-\tau\hat{V}}\,e^{O(\tau^2)}. \]

This motivates defining the transfer matrix as an approximated evolution operator for a single time step. The Trotter formula then reads \[ e^{-T\hat{H}} = \lim_{\substack{M\to\infty\\\tau=T/M}} \bigl[\hat{T}(\tau)\bigr]^M. \]

2.2 Symmetric Transfer Matrix

The simplest choice \(\hat{T}(\tau) = e^{-\tau\hat{K}}\,e^{-\tau\hat{V}}\) is not the only valid one. The Trotter formula also holds in a symmetric form, \[ e^{-T\hat{H}} = \lim_{M\to\infty} e^{\frac{\tau}{2}\hat{V}} \!\left(e^{-\tau\hat{V}}\,e^{-\tau\hat{K}}\,e^{\tau\hat{V}}\right)^{\!M} e^{-\frac{\tau}{2}\hat{V}} = \lim_{M\to\infty} \left(e^{-\frac{\tau}{2}\hat{V}}\,e^{-\tau\hat{K}}\,e^{-\frac{\tau}{2}\hat{V}}\right)^{\!M}, \] which leads to the symmetric transfer matrix \[ \hat{T}(\tau) \stackrel{\mathrm{def}}{=} e^{-\frac{\tau}{2}\hat{V}}\,e^{-\tau\hat{K}}\,e^{-\frac{\tau}{2}\hat{V}}. \] This choice satisfies the important properties \[ \hat{T}^\dagger = \hat{T}, \qquad \hat{T} \geq 0, \] which ensure that \(\hat{T}\) is diagonalizable with non-negative eigenvalues. These properties are not satisfied by the asymmetric choice \(e^{-\tau\hat{V}}\,e^{-\tau\hat{K}}\). All subsequent derivations use the symmetric transfer matrix.

2.3 Path-Integral Formula for Transition Amplitudes

From the Trotter formula, \[ \langle\varphi_T|e^{-T\hat{H}}|\varphi_0\rangle = \lim_{\substack{M\to\infty\\\tau=T/M}} \langle\varphi_T|\bigl[\hat{T}(\tau)\bigr]^M|\varphi_0\rangle. \] Expanding the \(M\)-fold product and inserting completeness relations \(\mathbf{1} = \int d\varphi_t\,|\varphi_t\rangle\langle\varphi_t|\) (with \(d\varphi_t = \prod_{\mathbf{x}}d\varphi(t,\mathbf{x})\)) at each intermediate time \(t = \tau, 2\tau, \ldots, T-\tau\), \[ \langle\varphi_T|\bigl[\hat{T}(\tau)\bigr]^M|\varphi_0\rangle = \int \left[\prod_{t=\tau,\ldots,T-\tau} \!\!d\varphi_t\right] \left[\prod_{t=0,\tau,\ldots,T-\tau} \!\!\langle\varphi_{t+\tau}|\hat{T}(\tau)|\varphi_t\rangle\right]. \] A temporal lattice emerges from this formula. Each factor \(\langle\varphi_{t+\tau}|\hat{T}(\tau)|\varphi_t\rangle\) can be computed analytically.

2.4 Matrix Element of the Transfer Matrix

We compute \(\langle\varphi_{t+\tau}|\hat{T}(\tau)|\varphi_t\rangle\) step by step.

Step ① — Definition of \(\hat{T}\): \[ \langle\varphi_{t+\tau}|\hat{T}(\tau)|\varphi_t\rangle = \langle\varphi_{t+\tau}|e^{-\frac{\tau}{2}\hat{V}}\,e^{-\tau\hat{K}}\,e^{-\frac{\tau}{2}\hat{V}}|\varphi_t\rangle. \]

Step ② — Since \(\hat{V} = V(\hat{\varphi})\) is diagonal in the field basis, \(e^{-\frac{\tau}{2}\hat{V}}|\varphi\rangle = e^{-\frac{\tau}{2}V(\varphi)}|\varphi\rangle\): \[ = e^{-\frac{\tau}{2}V(\varphi_{t+\tau})}\,e^{-\frac{\tau}{2}V(\varphi_t)} \underbrace{\langle\varphi_{t+\tau}|e^{-\tau\hat{K}}|\varphi_t\rangle}_{\text{Euclidean kernel}}. \]

Step ③ — Insert the momentum completeness relation \(\mathbf{1} = \int\!\left[\prod_{\mathbf{x}}\frac{d\pi(\mathbf{x})}{2\pi}\right]|\pi\rangle\langle\pi|\): \[ = e^{-\tau\frac{V(\varphi_{t+\tau})+V(\varphi_t)}{2}} \int\!\left[\prod_{\mathbf{x}}\frac{d\pi(\mathbf{x})}{2\pi}\right] \langle\varphi_{t+\tau}|e^{-\tau\hat{K}}|\pi\rangle\langle\pi|\varphi_t\rangle. \]

Step ④ — Use \(\hat{K}|\pi\rangle = \sum_{\mathbf{x}}\frac{a^3}{2}\pi^2(\mathbf{x})|\pi\rangle\) and \(\langle\pi|\varphi\rangle = \exp\!\left\{-i\sum_{\mathbf{x}}a^3\varphi(\mathbf{x})\pi(\mathbf{x})\right\}\): \[ = e^{-\tau\frac{V(\varphi_{t+\tau})+V(\varphi_t)}{2}} \int\!\left[\prod_{\mathbf{x}}\frac{d\pi(\mathbf{x})}{2\pi}\right] e^{-\tau\sum_{\mathbf{x}}\frac{a^3}{2}\pi^2(\mathbf{x})}\, e^{\,i\sum_{\mathbf{x}}a^3\pi(\mathbf{x})[\varphi_t(\mathbf{x})-\varphi_{t+\tau}(\mathbf{x})]}. \]

Step ⑤ — Factor over sites using \(e^{\sum_{\mathbf{x}}(\cdots)} = \prod_{\mathbf{x}}e^{(\cdots)}\), and define the forward time derivative \[ \partial_0^f\varphi_t(\mathbf{x}) \stackrel{\mathrm{def}}{=} \frac{\varphi_{t+\tau}(\mathbf{x})-\varphi_t(\mathbf{x})}{\tau}: \] \[ = e^{-\tau\frac{V(\varphi_{t+\tau})+V(\varphi_t)}{2}}\, \prod_{\mathbf{x}}\left\{\int\frac{d\pi}{2\pi}\, e^{-\frac{\tau a^3}{2}\pi^2 - i\tau a^3\pi\,\partial_0^f\varphi_t(\mathbf{x})}\right\}. \]

Step ⑥ — Evaluate the Gaussian integral \(\int_{-\infty}^{+\infty}dx\,e^{-\frac{\alpha}{2}x^2+ipx} = \sqrt{\frac{2\pi}{\alpha}}\,e^{-p^2/(2\alpha)}\) with \(\alpha = \tau a^3\), \(p = -\tau a^3\partial_0^f\varphi_t(\mathbf{x})\), \(x = \pi\): \[ = e^{-\tau\frac{V(\varphi_{t+\tau})+V(\varphi_t)}{2}}\, \prod_{\mathbf{x}}\!\left\{(2\pi\tau a^3)^{-1/2}\, e^{-\frac{\tau a^3}{2}[\partial_0^f\varphi_t(\mathbf{x})]^2}\right\} \] \[ = (2\pi\tau a^3)^{-N^3/2} \exp\!\left\{-\tau\!\left[\sum_{\mathbf{x}}\frac{a^3}{2}(\partial_0^f\varphi_t)^2 + \frac{V(\varphi_{t+\tau})+V(\varphi_t)}{2}\right]\right\}. \]

The expression in brackets is the Euclidean discretized Lagrangian: \[ L_E(t) \stackrel{\mathrm{def}}{=} \sum_{\mathbf{x}}\frac{a^3}{2}(\partial_0^f\varphi_t)^2 + \frac{V(\varphi_{t+\tau})+V(\varphi_t)}{2}. \]

2.5 Path-Integral Formula

Combining the results of the previous two sections, with the notation \([d\varphi]_{(0,T)} \stackrel{\mathrm{def}}{=} \prod_{t=\tau,\ldots,T-\tau} d\varphi_t\): \[ \langle\varphi_T|\bigl[\hat{T}(\tau)\bigr]^M|\varphi_0\rangle = (2\pi\tau a^3)^{-\frac{N^3M}{2}} \int[d\varphi]_{(0,T)}\, \exp\!\left\{-\sum_{t=0}^{T-\tau}\tau\,L_E(t)\right\}. \]

The exponent defines the discretized Euclidean action: \[ S^E_{[0,T)}(\varphi) \stackrel{\mathrm{def}}{=} \sum_{t=0}^{T-\tau}\tau\,L_E(t) = \sum_{t=0}^{T-\tau}\tau\!\left\{\sum_{\mathbf{x}}\frac{a^3}{2}(\partial_0^f\varphi_t)^2 + \frac{V(\varphi_{t+\tau})+V(\varphi_t)}{2}\right\}. \]

This is the path-integral formula for the transition amplitude between field eigenstates.

2.6 Summary

Hamiltonian (on 3d lattice): \[ \hat{H} = \sum_{\mathbf{x}} a^3\!\left[\frac{1}{2}\hat{\pi}^2 + \frac{1}{2}(\partial_k^f\hat{\varphi})^2 + \frac{m^2}{2}\hat{\varphi}^2 + \frac{1}{4!}\hat{\varphi}^4\right]\!(\mathbf{x}) = \hat{K} + \hat{V}. \]

Transfer matrix: \[ \hat{T}(\tau) = e^{-\frac{\tau}{2}\hat{V}}\,e^{-\tau\hat{K}}\,e^{-\frac{\tau}{2}\hat{V}}. \]

Discretized path-integral formula for transition amplitudes: \[ \langle\varphi_T|e^{-T\hat{H}}|\varphi_0\rangle = \lim_{\substack{M\to\infty\\\tau=T/M}} (2\pi\tau a^3)^{-\frac{N^3M}{2}} \int[d\varphi]_{(0,T)}\,e^{-S^E_{[0,T)}(\varphi)}. \]

Path-integral measure (over all fields on 4d lattice): \[ [d\varphi]_{(0,T)} = \prod_{t=\tau}^{T-\tau}\prod_{\mathbf{x}} d\varphi(t,\mathbf{x}). \]

Euclidean action (on 4d lattice): \[ S^E_{[0,T)}(\varphi) = \sum_{t=0}^{T-\tau}\tau\!\left\{ \sum_{\mathbf{x}}\frac{a^3}{2}(\partial_0^f\varphi)^2(t,\mathbf{x}) + \frac{V(\varphi_{t+\tau})+V(\varphi_t)}{2}\right\}. \]

The Hamiltonian \(\hat{H}\) is discretized on a 3d lattice with spacing \(a\) and periodic boundary conditions. The action \(S^E\) is discretized on a 4d lattice:

• 3d spatial lattice with spacing \(a\) and periodic boundary conditions;
• 1d temporal lattice with spacing \(\tau\) and Dirichlet boundary conditions (\(\varphi_0\) and \(\varphi_T\) are fixed).

A natural special choice is \(a = \tau\) (symmetric lattice).

2.7 Thermal Partition Function

The thermal partition function is \[ Z = \mathrm{tr}\,e^{-T\hat{H}} = \lim_{\substack{M\to\infty\\\tau=T/M}} \mathrm{tr}\bigl\{\hat{T}^M(\tau)\bigr\}. \] Note that \(T\) here plays the dual role of Euclidean time and inverse temperature: \(T = \beta\).

Using the cyclic property of the trace, \[ \mathrm{tr}\bigl\{\hat{T}^M(\tau)\bigr\} = \int d\varphi_0\,\langle\varphi_0|\hat{T}^M(\tau)|\varphi_0\rangle = \int d\varphi_0\,\langle\varphi_T|\hat{T}^M(\tau)|\varphi_0\rangle\Big|_{\varphi_T=\varphi_0}. \]

Applying the path-integral formula with the extended measure \([d\varphi]_{[0,T)} = \int d\varphi_0\cdot[d\varphi]_{(0,T)}\): \[ Z = (2\pi\tau a^3)^{-\frac{N^3M}{2}} \int[d\varphi]_{[0,T)}\,e^{-S^E_{[0,T)}(\varphi)}\Big|_{\varphi_T=\varphi_0}. \]

The key structural difference is:

• Matrix elements \(\langle\varphi_T|\hat{T}^M|\varphi_0\rangle\) correspond to Dirichlet boundary conditions in time.
• Partition function \(\mathrm{tr}\,\hat{T}^M\) corresponds to periodic boundary conditions in time.

When periodic boundary conditions are imposed (\(\varphi_T = \varphi_0\)), the two potential terms in \(L_E(t)\) contribute equally and the action simplifies: \[ S^E_{[0,T)}\big|_{\varphi_T=\varphi_0} = \sum_{t=0}^{T-\tau}\tau\!\left\{\sum_{\mathbf{x}}\frac{a^3}{2}(\partial_0^f\varphi_t)^2 + V(\varphi_t)\right\} = \sum_{t=0}^{T-\tau}\sum_{\mathbf{x}}\tau a^3\!\left[\frac{1}{2}(\partial_0^f\varphi)^2 + \frac{m^2}{2}\varphi^2 + \frac{1}{4!}\varphi^4\right]. \]

2.8 Core Concepts of Thermal QFT

A quantum mechanical system in thermal equilibrium with inverse temperature \(\beta\) is described by the statistical state (density matrix) \[ \hat{\rho} = \frac{1}{Z}\,e^{-\beta\hat{H}}, \qquad Z = \mathrm{tr}\,e^{-\beta\hat{H}} = \text{partition function} \qquad [\text{canonical ensemble}]. \]

The thermal expectation value of a generic observable \(\hat{A}\) is \[ \langle\hat{A}\rangle_\beta = \mathrm{tr}\{\hat{\rho}\hat{A}\} = \frac{1}{Z}\mathrm{tr}\!\left\{e^{-\beta\hat{H}}\hat{A}\right\}. \]

The inverse temperature, the Euclidean time, and the temporal extent of the lattice are all identified: \(\beta = T\).

The thermal \(n\)-point functions are defined as \[ \mathrm{tr}\bigl\{\hat{\rho}\,\hat{\varphi}(t_n,\mathbf{x}_n)\cdots\hat{\varphi}(t_1,\mathbf{x}_1)\bigr\}, \] where \(\hat{\varphi}(t,\mathbf{x}) = e^{t\hat{H}}\hat{\varphi}(\mathbf{x})e^{-t\hat{H}}\) is the field in the Euclidean–Heisenberg picture. Expanding the density matrix, \[ = \frac{1}{Z}\mathrm{tr}\!\left\{ e^{-(T-t_n)\hat{H}}\hat{\varphi}(\mathbf{x}_n)\, e^{-(t_n-t_{n-1})\hat{H}}\hat{\varphi}(\mathbf{x}_{n-1})\cdots \hat{\varphi}(\mathbf{x}_2)\,e^{-(t_2-t_1)\hat{H}}\hat{\varphi}(\mathbf{x}_1)\, e^{-t_1\hat{H}}\right\}. \] We will primarily consider time-ordered \(n\)-point functions, where \(T > t_n > t_{n-1} > \cdots > t_2 > t_1 > 0\).

2.9 Path-Integral Formula for Thermal \(n\)-Point Functions

We derive the path-integral representation of the thermal time-ordered \(n\)-point function. Let \(M = T/\tau\) be an integer. Since \(t_k/\tau\) is generally not an integer, each \(t_k\) is approximated by a nearby temporal lattice site \(\tilde{t}_k\) (the tilde will be dropped henceforth).

Replace each factor \(e^{-\Delta t\,\hat{H}}\) by the appropriate power \(\hat{T}(\tau)^{\Delta t/\tau}\) of the transfer matrix: \[ \mathrm{tr}\!\left\{\hat{\rho}\,\hat{\varphi}(t_n,\mathbf{x}_n)\cdots\hat{\varphi}(t_1,\mathbf{x}_1)\right\} = \lim_{\substack{M\to\infty\\\tau=T/M}} \frac{ \mathrm{tr}\!\left\{ \hat{T}(\tau)^{\frac{T-t_n}{\tau}}\hat{\varphi}(\mathbf{x}_n)\, \hat{T}(\tau)^{\frac{t_n-t_{n-1}}{\tau}}\cdots \hat{\varphi}(\mathbf{x}_1)\,\hat{T}(\tau)^{\frac{t_1}{\tau}} \right\} }{ \mathrm{tr}\!\left\{\hat{T}(\tau)^{T/\tau}\right\} }. \]

At each field insertion, a completeness relation \(\mathbf{1} = \int d\varphi_{t_k}\,|\varphi_{t_k}\rangle\langle\varphi_{t_k}|\) is inserted, so that \[ \hat{\varphi}(\mathbf{x}_k) = \int d\varphi_{t_k}\,|\varphi_{t_k}\rangle\,\varphi(\mathbf{x}_k)\langle\varphi_{t_k}|. \] After the same Gaussian integration as before, the factor \((2\pi\tau a^3)^{-N^3M/2}\) cancels between numerator and denominator, giving the path-integral formula for thermal time-ordered \(n\)-point functions: \[ \mathrm{tr}\!\left\{\hat{\rho}\,\hat{\varphi}(t_n,\mathbf{x}_n)\cdots\hat{\varphi}(t_1,\mathbf{x}_1)\right\} = \frac{ \displaystyle\int[d\varphi]_{[0,T)}\,e^{-S^E_{[0,T)}(\varphi)} \prod_{k=1}^n\varphi(t_k,\mathbf{x}_k)\Big|_{\varphi_T=\varphi_0} }{ \displaystyle\int[d\varphi]_{[0,T)}\,e^{-S^E_{[0,T)}(\varphi)}\Big|_{\varphi_T=\varphi_0} } \equiv \left\langle\prod_{k=1}^n\varphi(t_k,\mathbf{x}_k)\right\rangle, \] where the right-hand side is the path-integral expectation value on an \(M\times N^3\) lattice with spacings \(\tau, a\) and periodic boundary conditions in space and time.

Probabilistic interpretation. The path-integral expectation value can be written as \[ \langle A(\varphi)\rangle = \int[d\varphi]\,A(\varphi)\,P(\varphi), \qquad P(\varphi) = \frac{e^{-S(\varphi)}}{\displaystyle\int[d\varphi]\,e^{-S(\varphi)}}. \] When \(S(\varphi)\) is real, one has \(P(\varphi) \geq 0\) and \(\int[d\varphi]\,P(\varphi) = 1\): the weight \(P(\varphi)\) is a probability distribution on the space of fields on the 4d lattice, and the path-integral expectation value is the expectation value with respect to \(P(\varphi)\).

2.10 Summary (2)

The full framework, from the QFT Hamiltonian to path-integral observables, is collected here.

QFT Hamiltonian (continuum): \[ \hat{H}_{QFT} = \int d^3x\left[\frac{1}{2}\hat{\pi}^2 + \frac{1}{2}(\partial_k\hat{\varphi})^2 + \frac{m^2}{2}\hat{\varphi}^2 + \frac{1}{4!}\hat{\varphi}^4\right]. \]

Hamiltonian on 3d lattice: \[ \hat{H} = \sum_{\mathbf{x}} a^3\!\left[\frac{1}{2}\hat{\pi}^2 + \frac{1}{2}(\partial_k^f\hat{\varphi})^2 + \frac{m^2}{2}\hat{\varphi}^2 + \frac{1}{4!}\hat{\varphi}^4\right] = \hat{K}+\hat{V}. \]

Transfer matrix: \(\hat{T}(\tau) = e^{-\frac{\tau}{2}\hat{V}}\,e^{-\tau\hat{K}}\,e^{-\frac{\tau}{2}\hat{V}}\).

Euclidean action on 4d lattice (for any time interval \(I\)): \[ S_I(\varphi) = \sum_{t\in I}\tau\!\left\{ \sum_{\mathbf{x}}\frac{a^3}{2}(\partial_0^f\varphi)^2(t,\mathbf{x}) + \frac{V(\varphi_t)+V(\varphi_{t+\tau})}{2}\right\}. \]

Path-integral measure on 4d lattice: \([d\varphi]_I = \prod_{t\in I}\prod_{\mathbf{x}}d\varphi(t,\mathbf{x})\).

	Matrix elements of Euclidean evolution operator	Thermal EV of time-ordered product of fields
Continuous time	\(\langle\varphi_T\lvert e^{-T\hat{H}}\rvert\varphi_0\rangle\)	\(\frac{1}{Z}\mathrm{tr}\!\left\{e^{-T\hat{H}}\hat\varphi(t_n,\mathbf{x}_n)\cdots\hat\varphi(t_1,\mathbf{x}_1)\right\}\), \(\;T>t_n>\cdots>t_1>0\)
Discrete time	\(\langle\varphi_T\lvert\hat{T}^M\rvert\varphi_0\rangle\)	\(\frac{1}{Z_{\mathrm{disc}}}\mathrm{tr}\!\left\{\hat{T}^{\frac{T-t_n}{\tau}}\hat\varphi(\mathbf{x}_n)\cdots\hat\varphi(\mathbf{x}_1)\hat{T}^{\frac{t_1}{\tau}}\right\}\)
Path-integral	\((2\pi\tau a^3)^{-\frac{N^3M}{2}}\int[d\varphi]_{(0,T)}\,e^{-S_{[0,T)}}\)	\(\frac{\int[d\varphi]_{[0,T)}\,e^{-S_{[0,T)}}\prod_{k=1}^n\varphi(t_k,\mathbf{x}_k)\big\vert_{\varphi_T=\varphi_0}}{\int[d\varphi]_{[0,T)}\,e^{-S_{[0,T)}}\big\vert_{\varphi_T=\varphi_0}}\)

2.11 The QFT Limit(s)

The discretization introduced four artificial parameters:

Parameter	Meaning	Related physical scale
\(a\)	spatial lattice spacing	\(L = aN\) = spatial extent
\(\tau\)	temporal lattice spacing	\(T = \tau M\) = temporal extent
\(N\)	number of sites per spatial direction
\(M\)	number of temporal sites

QFT is recovered when \(a, \tau \to 0\) and \(N, M \to \infty\). The limit \(a \to 0\) requires renormalization (to be discussed later). These limits can be taken in several ways:

\[ \lim_{\substack{a,\tau\to 0\\N,M\to\infty\\a/\tau\;\text{fixed}}}[\text{lattice obs.}] \;\longrightarrow\; \text{QFT} \quad (T, L = \infty), \]

\[ \lim_{\substack{a,\tau\to 0\\N,M\to\infty\\L,T\;\text{fixed}}}[\text{lattice obs.}] \;\longrightarrow\; \text{finite-}T\text{ finite-}L\text{ QFT}, \]

\[ \lim_{\substack{a,\tau\to 0\\T\;\text{fixed}}}\;\lim_{N\to\infty}[\text{lattice obs.}] \;\longrightarrow\; \text{finite-}T\text{ QFT} \quad (L = \infty), \]

\[ \lim_{\substack{a,\tau\to 0\\L\;\text{fixed}}}\;\lim_{M\to\infty}[\text{lattice obs.}] \;\longrightarrow\; \text{finite-}L\text{ QFT} \quad (T = \infty). \]

2.12 Observables

A central question is: which quantities are physically meaningful and computable on the lattice? Thinking of typical theories (\(\varphi^4\), QED, QCD, the Standard Model, …):

• Experimentally measurable quantities include cross sections involving stable or long-lived asymptotic particles (e.g. the muon \(\mu\)).
• Theoretically well defined quantities are those free from renormalization scheme dependence, infrared divergences, and gauge dependence.
• Theoretically ambiguous quantities are affected by these issues and require additional care.

To make progress we must answer: what is a particle in QFT, and what is its mass? The word “particle” is used for many different objects in physics. In this course we focus on stable asymptotic particles (not necessarily elementary): electrons, photons, protons, stable nuclei, stable atoms, etc. Cross sections and masses of these particles are both experimentally accessible and theoretically well defined.