Learning objectives / 学习目标
- 在有限周期盒中把 free field 展开为 Fourier modes，并把 Hamiltonian 识别为独立 harmonic oscillators；
- 定义 ladder operators、commutators 与 Fock states，计算 occupation number、energy 与 momentum；
- 解释 particle 是 quantum field excitation 的有限能态，并说明 localized wave packet、interaction 与 observer dependence 的边界；
- 区分 vacuum expectation、vacuum fluctuation、zero-point divergence 与可测 difference/correlation；
- 把 regulator、renormalized parameter、running coupling、renormalization scale 与 effective-field-theory truncation 组织为预测流程。
Prerequisites / 先修知识
- Classical Fields and Symmetry：field action、canonical momentum、normal modes、Noether charges 与 gauge redundancy。
- Quantum Harmonic Oscillator：\([\hat q,\hat p]=i\hbar\)、ladder operators 与 \(E_n=\hbar\omega(n+1/2)\)。
- Dispersion and Wave Packets：Fourier modes、group velocity 与 localization。
- Four-Vectors：energy–momentum relation 与 Lorentz covariance。
Scope and conventions / 范围与约定
- 主线是 free real scalar field 的 canonical quantization；fermionic anticommutators、interacting gauge-field constraints 与 path integral 只作接口。
- 先放在 finite periodic box，令 mode set 离散且数学可控；continuum/volume limit 最后取。盒子不是宇宙真实边界。
- 推导保留 \(\hbar\) 以连接 SI quantum oscillator；relativistic dispersion 处可用 \(\hbar=c=1\)，每次声明。
- Particle concept 对 free field 和 asymptotic states 最清楚；interacting、curved spacetime、accelerated observers 下不必有唯一全局 particle number。
- Vacuum 是给定 Hamiltonian/representation 的 lowest-energy state，不是“完全没有 field”或 classical zero configuration。
- Renormalization 不是用 infinity 任意减 infinity；regulator 只是 intermediate definition，physical predictions 经 measured inputs fixed 后应在目标精度内对 regulator insensitive。
Notation and modeling contract / 符号与建模契约
| Symbol | Meaning | Unit/convention |
|---|---|---|
| \(q_{\mathbf k},p_{\mathbf k}\) | canonical amplitude and momentum of mode \(\mathbf k\) | normalization-dependent |
| \(\omega_{\mathbf k}\) | mode angular frequency | \(\mathrm{s^{-1}}\) or energy in natural units |
| \(\hat a_{\mathbf k},\hat a^\dagger_{\mathbf k}\) | annihilation/creation operators | 1 under standard normalization |
| \(\hat N_{\mathbf k}\) | \(\hat a^\dagger_{\mathbf k}\hat a_{\mathbf k}\) | 1 |
| \(\lvert n_{\mathbf k}\rangle\) | mode occupation eigenstate | normalized state |
| \(\lvert0\rangle\) | vacuum for stated free Hamiltonian | state |
| \(\Lambda\) | ultraviolet regulator/cutoff | energy or inverse length |
| \(\mu_R\) | renormalization scale | energy |
| \(g(\mu_R)\) | renormalized running coupling | model-dependent |
Dagger means Hermitian adjoint。\(\mu_R\) 专指 renormalization scale，避免与 string linear density 或 signal strength \(\mu\) 混淆。
1. Field modes as harmonic oscillators / 场模作为谐振子
Knowledge ID: P11-U05-L02-K01
先把 linear classical field 放在 volume \(V\) 的 periodic box。Fourier expansion 把 spatial coupling diagonalize；real field 要满足 \(q_{-\mathbf k}=q_{\mathbf k}^*\)，独立 real modes 可用 sine/cosine basis 计数。令 \(\mathcal I\) 表示这种独立 real normal-mode set：零模只计一次，每一对 \(\pm\mathbf k\) 改写成独立 cosine/sine amplitudes 后各计一次。Free-field Hamiltonian 经过适当 normalization 成为
\[H=\sum_{\mathbf k\in\mathcal I} \left[ \frac12p_{\mathbf k}^2+\frac12\omega_{\mathbf k}^2q_{\mathbf k}^2 \right].\]
这里每个 \(\mathbf k\in\mathcal I\) 是一个独立 harmonic oscillator 标签，而不是把 full signed complex coefficients \(q_{\mathbf k}\) 与 \(q_{-\mathbf k}\) 当成两个独立自由度，更不是一个 classical particle。Relativistic scalar mode 有
\[\omega_{\mathbf k}^2=c^2|\mathbf k|^2+\frac{m^2c^4}{\hbar^2}.\]
Massless field 取 \(\omega=c|\mathbf k|\)。Boundary conditions 离散化 \(\mathbf k\)；box 长度 \(L_i\) 上 \(k_i=2\pi n_i/L_i\)。若改用 conventional traveling-wave operator expansion，signed \(\mathbf k\) 的 \(a_{\mathbf k}\) labels particles of different momenta；这与把 related classical coefficients \(q_{\mathbf k}\)、\(q_{-\mathbf k}\) 独立计数不是一回事。Continuum replacement 及其 measure 必须与所选 independent-mode/normalization convention 配套，不能机械写一个 full-space integral 后再重复乘二。
Interacting theory 会耦合 modes，不能继续把每个 mode 视为 independent oscillator；free decomposition 仍提供 perturbation theory 的 basis。Gauge fields 还有 constraints/physical polarization projection，fermion fields 使用 anticommutators，因此本课的 oscillator picture 是桥梁而非所有细节。
图 1 — Fourier decomposition 把一个 field configuration 写成 normal-mode amplitudes；quantization 作用于这些 canonical modes。
图 2 — 每个 free mode 的 Hamiltonian 是 oscillator；水平线表示量子化后等间隔能级，不是不同空间位置。
2. Creation and annihilation operators / 产生与湮灭算符
Knowledge ID: P11-U05-L02-K02
对每个 bosonic mode 定义
\[\hat a_{\mathbf k}= \sqrt{\frac{\omega_{\mathbf k}}{2\hbar}}\hat q_{\mathbf k} +\frac{i}{\sqrt{2\hbar\omega_{\mathbf k}}}\hat p_{\mathbf k},\]
\[\hat a_{\mathbf k}^\dagger= \sqrt{\frac{\omega_{\mathbf k}}{2\hbar}}\hat q_{\mathbf k} -\frac{i}{\sqrt{2\hbar\omega_{\mathbf k}}}\hat p_{\mathbf k}.\]
Canonical commutator gives
\[[\hat a_{\mathbf k},\hat a_{\mathbf k'}^\dagger] =\delta_{\mathbf k\mathbf k'}, \qquad \hat H=\sum_{\mathbf k}\hbar\omega_{\mathbf k} \left(\hat N_{\mathbf k}+\frac12\right).\]
Vacuum satisfies \(\hat a_{\mathbf k}|0\rangle=0\)。Fock states
\[|n_{\mathbf k}\rangle=\frac{(\hat a_{\mathbf k}^\dagger)^n}{\sqrt{n!}}|0\rangle\]
obey \(\hat N|n\rangle=n|n\rangle\)，\(\hat a|n\rangle=\sqrt n|n-1\rangle\)，\(\hat a^\dagger|n\rangle=\sqrt{n+1}|n+1\rangle\)。Square roots preserve normalization。For fermionic modes, anticommutators imply \(n=0,1\)；直接复用 bosonic ladder algebra 会违反 Pauli exclusion。
图 3 — Bosonic ladder operators 在同一 mode 的 Fock levels 间移动；系数 \(\sqrt{n+1}\)、\(\sqrt n\) 未以箭长编码。
图 4 — 多 mode Fock space 以 occupation tuple 标记；一个 quantum state 可含多个 modes 的不同 occupation。
3. Particles as field excitations / 粒子作为场的激发
Knowledge ID: P11-U05-L02-K03
Free relativistic field 的 one-quantum state \(\hat a_{\mathbf k}^\dagger|0\rangle\) 有 energy \(\hbar\omega_{\mathbf k}\) 与 momentum \(\hbar\mathbf k\)，满足
\[E_{\mathbf k}^2=p_{\mathbf k}^2c^2+m^2c^4.\]
这就是“particle as field excitation”：field 是 primary dynamical object，particle 是其量子态的 excitation pattern。Momentum eigenstate 遍布全空间；localized particle 要用 wave packet
\[|\Psi\rangle=\int d^3k\,f(\mathbf k)\hat a_{\mathbf k}^\dagger|0\rangle, \qquad \int d^3k\,|f(\mathbf k)|^2=1\]
（normalization convention simplified）。Localization 与 momentum spread 互为 Fourier tradeoff。
In an interacting theory，bare mode occupation generally not conserved；interactions can create/annihilate quanta while Noether charges and four-momentum remain conserved。Particle interpretation 最清楚地属于 long-lived asymptotic states and detector excitations。Unstable resonances 由 pole/line shape 与 decay products 定义，不是永远存在的 Fock particle；confinement 又使 quark/gluon modes 不作为 isolated asymptotic particles。Curved spacetime/accelerated observers may disagree on positive frequency and particle content，因此“particle number”不是比 field 更 fundamental 的 universal observable。
图 5 — Localized excitation 需要多 \(\mathbf k\) modes 的 superposition；包络宽度与 momentum spread 互制。
图 6 — Interactions 可改变 mode occupations/particle count；图只表达 quantum numbers 与 four-momentum bookkeeping，不是 classical paths。
4. Vacuum fluctuations / 真空涨落
Knowledge ID: P11-U05-L02-K04
Vacuum \(|0\rangle\) has no free-particle occupation，but each oscillator has zero-point energy \(\frac12\hbar\omega_{\mathbf k}\) and nonzero quadrature variance。For a free scalar field，often
\[\langle0|\hat\phi(x)|0\rangle=0, \qquad \langle0|\hat\phi(x)\hat\phi(y)|0\rangle\ne0.\]
The two-point function encodes fluctuations, propagation and correlations；operator ordering and spacetime separation matter。Vacuum fluctuation does not mean a classical random field with simultaneously definite values，nor particles “borrowing energy” in violation of conservation。Energy-time uncertainty is not a license for temporary energy nonconservation。
Formal zero-point sum
\[E_0=\frac12\sum_{\mathbf k}\hbar\omega_{\mathbf k}\]
diverges in continuum。A regulator makes expressions defined；observable questions often involve differences, forces, transition rates or renormalized parameters。Casimir force is commonly described through changed mode spectrum/boundary-dependent vacuum energy, but a complete calculation must include material response and subtract boundary-independent pieces。Vacuum state also depends on Hamiltonian/background and observer; “vacuum is empty space with a measurable universal energy density equal to a raw mode sum” is not a justified inference。
图 7 — Oscillator vacuum wavefunction has nonzero width and zero-point energy；shaded region is qualitative, not the exact Gaussian。
图 8 — Boundaries 改变 allowed correlations/modes，产生可测 force difference；图不声称真空中存在 classical standing waves。
5. Renormalization as scale-dependent description / 重整化作为标度相关描述
Knowledge ID: P11-U05-L02-K05
无限 modes 使 loop integrals 和 local operator products 对 high momentum 敏感。Regularization（cutoff、dimensional、lattice 等）先定义 intermediate expressions。Write bare parameters as regulator-dependent combinations of measured renormalized parameters and counterterms，then impose renormalization conditions。最终 finite observable 在目标 perturbative/order precision 内不应依 arbitrary regulator。
Renormalized coupling depends on scale：
\[\beta(g)=\mu_R\frac{dg}{d\mu_R}.\]
This running is physical scale dependence after a scheme and input are specified，not an experimenter随意 changing a constant。Renormalization scale \(\mu_R\) separates calculation organization；exact result is scale independent，truncated perturbation retains residual \(\mu_R\) dependence used as one diagnostic, not a universal uncertainty theorem。
Effective field theory（EFT）写出与 symmetries compatible 的 local operators，按 powers of \(E/\Lambda\) 排序：
\[\mathcal L_{\mathrm{EFT}} =\mathcal L_{\mathrm{low}} +\sum_i\frac{C_i(\mu_R)}{\Lambda^{d_i-4}}\mathcal O_i.\]
At \(E\ll\Lambda\) higher-dimension terms suppressed；coefficients carry short-distance physics and run with scale。Renormalization 因此是 scale-to-scale information bookkeeping：measure finite set of inputs，predict other observables with controlled truncation。If \(E\) approaches \(\Lambda\) or omitted operators are unsuppressed，EFT expansion fails rather than becoming “slightly less exact”。
图 9 — UV regulator 暂时限制高-\(k\) contributions；physical prediction 需 renormalization 后检查 regulator independence。
图 10 — Renormalization-group flow 连接不同 scales 的参数；曲线方向是教学示意，具体 beta function 决定上升或下降。
Worked Examples / 完整例题
Worked Example 1 — 周期盒中的 field mode
已知： 1D massless wave field，periodic length \(L=2.00\,\mathrm m\)，wave speed \(v=3.00\,\mathrm{m\,s^{-1}}\)，mode index \(n=2\)。
系统/坐标： \(0\le x<L\)，\(u(x+L)=u(x)\)；linear free field。
模型理由： periodic Fourier mode \(k_n=2\pi n/L\)，\(\omega_n=v\lvert k_n\rvert\)。
\[k_2=\frac{4\pi}{2.00}=6.28\,\mathrm{m^{-1}}, \qquad \omega_2=(3.00)(6.28)=18.8\,\mathrm{s^{-1}}.\]
\(f_2=\omega_2/(2\pi)=3.00\,\mathrm{Hz}\)。检查： wavelength \(L/n=1.00\,\mathrm m\)，\(v/\lambda=3.00\,\mathrm{Hz}\) 一致。Quantization 后相邻 occupation levels 相差 \(\hbar\omega_2\)；盒子只做 mode discretization。
Worked Example 2 — Ladder action 与 normalization
已知： 一个 bosonic mode 处于 \(|3\rangle\)，frequency \(\omega\)；standard ladder normalization。
系统： 单 free harmonic mode；phase/reference frame 不影响 occupation。
\[\hat a|3\rangle=\sqrt3|2\rangle, \qquad \hat a^\dagger|3\rangle=\sqrt4|4\rangle=2|4\rangle.\]
Applying number operator gives \(\hat N|3\rangle=3|3\rangle\)；energy \(E_3=\frac72\hbar\omega\)。检查： Lowered state norm is \(3\) before coefficient normalization，consistent with \(\langle3|a^\dagger a|3\rangle=3\)。\(a\) is not an energy-losing classical force；it maps states in Hilbert space。
Worked Example 3 — Two-mode Fock energy and momentum
已知： periodic free bosonic field has traveling modes \(\mathbf k\) and \(-\mathbf k\) with equal \(\omega_k\)，state \(|n_{\mathbf k}=2,n_{-\mathbf k}=1\rangle\)。
reference frame/system： box rest frame；include only excitation energy above vacuum。
\[E_{\mathrm{exc}}=(2+1)\hbar\omega_k=3\hbar\omega_k,\]
\[\mathbf P=(2\hbar\mathbf k)+(1)(-\hbar\mathbf k)=\hbar\mathbf k.\]
Total occupation is 3, but net momentum is one quantum because opposite modes partly cancel。检查： Energy adds positive mode energies while momentum is vectorial。Interactions may change occupation but must conserve total four-momentum in a translation-invariant closed system。
Worked Example 4 — Vacuum oscillator variance in SI
已知： effective oscillator mass \(M=1.00\,\mathrm{kg}\)、\(\omega=4.00\,\mathrm{s^{-1}}\)，ground state；\(\hbar=1.055\times10^{-34}\,\mathrm{J\,s}\)。
系统： 一个 normalized mechanical mode（用于理解，不是直接给 field-at-a-point variance）。
\[\langle0|q^2|0\rangle=\frac{\hbar}{2M\omega} =\frac{1.055\times10^{-34}}{8.00} =1.32\times10^{-35}\,\mathrm{m^2}.\]
\[q_{\mathrm{rms}}=\sqrt{\langle q^2\rangle}=3.63\times10^{-18}\,\mathrm m.\]
量纲检查： \(\mathrm{(J\,s)/(kg\,s^{-1})=m^2}\)。Mean \(\langle q\rangle=0\) 仍可有 nonzero variance；这不表示 ground state 被 classical random kicks 驱动。
Worked Example 5 — Toy one-loop running
已知： 教学 beta function \(dg/d\ln\mu_R=b g^3\)，\(g(\mu_0)=0.300\)，\(b=0.500\)，\(\mu_1=10\mu_0\)。
模型： dimensionless toy coupling、one-loop、no thresholds；不是 Standard Model numerical prediction。
Integrating gives
\[\frac{1}{g^2(\mu_1)}=\frac{1}{g^2(\mu_0)}-2b\ln\frac{\mu_1}{\mu_0}.\]
\[\frac1{g^2(\mu_1)}=11.111-2.303=8.808, \qquad g(\mu_1)=0.337.\]
检查： Positive \(b\) makes \(g\) grow with scale。If denominator approached zero, one-loop perturbation would fail；it would not justify extrapolating past the pole。
Lesson synthesis / 全课综合
Free-field quantization is a controlled sequence：finite box makes modes discrete；Fourier transform turns each normal mode into a canonical oscillator；commutators and ladder operators produce Fock occupations；one-quantum states carry relativistic energy–momentum and are called particles；vacuum is the joint ground state with nonzero correlations。Infinite modes then expose ultraviolet sensitivity，so regulator、measured renormalized inputs、running parameters and EFT power counting organize finite predictions across scales。The field—not a permanently fixed particle count—is the common object linking creation、annihilation、vacuum and interaction。
Formula-and-decision table / 公式与决策表
| Task | Relation | Conditions | Do not infer |
|---|---|---|---|
| decompose free field | \(H=\sum_{\mathbf k\in\mathcal I}(p_{\mathbf k}^2+\omega_{\mathbf k}^2q_{\mathbf k}^2)/2\) | \(\mathcal I\) is an independent real-mode set | signed Fourier coefficients are all independent |
| quantize bosonic mode | \([a_k,a_{k'}^\dagger]=\delta_{kk'}\) | canonical normalization | apply to fermions |
| occupation energy | \(H_k=\hbar\omega_k(N_k+1/2)\) | free oscillator mode | particle number always conserved |
| make one quantum | \(\lvert1_k\rangle=a_k^\dagger\lvert0\rangle\) | stated vacuum and mode basis | localized point particle |
| vacuum correlation | \(\langle0\rvert\phi(x)\phi(y)\lvert0\rangle\) | operator ordering/state specified | classical hidden field values |
| running | \(\beta(g)=\mu_Rdg/d\mu_R\) | scheme/model/order stated | arbitrary variation of nature |
| EFT expansion | \(\mathcal L=\mathcal L_{low}+\sum C_i\mathcal O_i/\Lambda^{d_i-4}\) | \(E/\Lambda\ll1\) and symmetries fixed | valid near/above cutoff |
Model boundaries / 模型边界
- Mode-as-oscillator decomposition is exact for free linear fields after handling reality, polarization and constraints；interactions couple modes。
- Bosonic commutators and unbounded occupation do not apply to fermionic modes；they use anticommutators and \(0/1\) occupation。
- A momentum eigenstate is delocalized；localized particle requires a packet and detector model。
- Particle number can be basis-, interaction- and observer-dependent；conserved Noether charges are conceptually distinct from occupation number。
- Vacuum fluctuations are state/operator correlations, not temporary violations of energy conservation or classical particles popping in and out。
- Raw zero-point sums are regulator-dependent；only consistently renormalized observables or differences support physical claims。
- Running couplings depend on scheme and scale，while observables do not when calculated exactly；truncation leaves controlled residual dependence。
- EFT must state expansion parameter and cutoff；renormalizability in the older power-counting sense is not the same as fundamental validity to unlimited energy。
Misconception clinic / 误区诊所
- “Quantizing a field means chopping it into little particles。” It promotes mode amplitudes to operators；particle excitations emerge as states。
- “Creation operator creates a tiny object at a point。” \(a_k^\dagger\) creates a momentum-mode quantum；localization needs superposition。
- “Vacuum has zero particles, so every observable is zero。” One-point mean may vanish while variance/two-point correlation does not。
- “Virtual particles borrow energy for \(\Delta t\sim\hbar/\Delta E\)。” Perturbative internal lines are bookkeeping contributions；exact theory conserves energy under time translation。
- “Renormalization hides infinities by arbitrary subtraction。” Measured conditions fix parameters，and regulator-independent predictions test the procedure。
Practice and retrieval / 练习与提取
- [K01·mode] Periodic length \(L\) 的 allowed 1D wave numbers 是什么？\(n=0\) mode 有何特殊？
- [K01·边界] 为什么 real field 的 \(q_k\) 与 \(q_{-k}\) 不是独立 complex variables？
- [K02·计算] 求 \(a|5\rangle\)、\(a^\dagger|5\rangle\) 与 \(N|5\rangle\)。
- [K02·比较] Bosonic 与 fermionic occupation 的 algebra 核心差别是什么？
- [K03·概念] 为什么 \(a_k^\dagger|0\rangle\) 不表示 particle 已定位在某个 \(x\)？
- [K03·守恒] Interactions 可改变 particle number，为什么不违反 conservation laws？
- [K04·判断] \(\langle0|\phi|0\rangle=0\) 是否推出 \(\langle0|\phi^2|0\rangle=0\)？
- [K04·边界] Casimir effect 是否等于直接测量 \(\frac12\sum_k\hbar\omega_k\) 的绝对值？
- [K05·概念] Regulator 与 renormalization scale 是同一个东西吗？
- [K01–K05·综合] 写出 free scalar QFT 从 classical action 到 finite prediction 的七步链。
第 1 题答案
\(k_n=2\pi n/L\)。\(n=0\) is spatially uniform；its frequency/constraint depends on mass and boundary dynamics，不能自动删除。
第 2 题答案
Reality requires \(\phi^*=\phi\)，hence Fourier coefficients satisfy \(q_{-k}=q_k^*\)；independent real sine/cosine modes avoid double counting。
第 3 题答案
\(a|5\rangle=\sqrt5|4\rangle\)，\(a^\dagger|5\rangle=\sqrt6|6\rangle\)，\(N|5\rangle=5|5\rangle\)。
第 4 题答案
Bosons use commutators and \(n=0,1,2,\ldots\)；fermions use anticommutators，\((a^\dagger)^2=0\) enforces \(n=0,1\) per mode。
第 5 题答案
It is momentum eigenmode with spatially extended phase。A localized state needs superposition \(\int f(k)a_k^\dagger|0\rangle\) and has momentum spread。
第 6 题答案
Particle occupation is not generally a Noether charge。Interaction vertices can change occupations while conserving total energy–momentum and exact internal charges。
第 7 题答案
No。Like a symmetric ground-state wavefunction, mean can vanish while variance/two-point function is positive/nonzero。
第 8 题答案
No。Observable is a force/energy difference after consistent subtraction and material/boundary response；raw absolute continuum sum is divergent and regulator-dependent。
第 9 题答案
No。Regulator makes intermediate UV expressions defined；\(\mu_R\) labels renormalized parameter convention/scale。Physical prediction should lose regulator dependence but couplings run with \(\mu_R\)。
第 10 题答案
Derive modes/dispersion；canonical variables；promote to operators；impose commutators；define vacuum/Fock states；compute correlations/amplitudes with regulator；fix renormalized inputs and verify regulator/scale/truncation control。
Exit ticket / 离堂检验
- 用“mode—operator—state—detector”四层语言解释一粒 photon/one scalar quantum，避免 classical particle picture。
- 对“vacuum fluctuation 让粒子短暂违反能量守恒”写出一段纠错，至少提到 Hamiltonian state、correlation 与 perturbative bookkeeping。
References and source notes / 参考资料与来源说明
以下链接均于 2026-08-30 实际打开并核对；正文重新组织，所有图均依据方程原创绘制。
- David Tong — Lectures on Quantum Field Theory — canonical quantization, free fields, particles, vacuum and renormalization overview
- David Tong — Free Fields — oscillator modes, commutators, Fock states, vacuum and particle interpretation
- MIT OCW 8.323 — Relativistic Quantum Field Theory I — canonical scalar quantization, particles, propagators, Maxwell field and renormalization
- MIT OCW 8.324 — Relativistic Quantum Field Theory II — field/mass renormalization, gauge quantization and renormalization-group structure
- University of Hamburg — Quantum Field Theory I lecture notes — scalar quantization, Fock space, QED and dimensional/field/charge renormalization
Source-to-content map
| Source | Used for | Knowledge ID / Section | Model boundary or convention |
|---|---|---|---|
| Tong QFT | course-level bridge from classical fields to quantization/renormalization | K01–K05 | graduate source; this lesson uses only conceptual and free-field core |
| Tong Free Fields | mode oscillators, Fock algebra, vacuum and particles | K01–K04 | normalization conventions are stated/simplified |
| MIT 8.323 | canonical quantization through quantum fluctuations and renormalization | K01–K05 | free scalar is the main model; gauge/fermion details deferred |
| MIT 8.324 | field/mass renormalization and RG | K05 | advanced formulas serve boundary/reference, not rote extension |
| Hamburg QFT notes | scalar modes, Fock space and renormalized QED sequence | K01, K02, K05 | metric and normalization conventions unified here |
复习与拓展
- 跨 Part 下一课 Coordinates, Time, and Angular Measurement 转向观测宇宙；QFT particles and vacuum remain the microphysical basis of photons and detectors。
- 回链 Classical Fields and Symmetry：mode frequencies、canonical momenta 与 Noether charges 是 quantization 的输入。
- 回链 Gauge Fields and Quantum Electrodynamics：Feynman diagrams、running coupling 与 vacuum polarization 都建立在本课的 operator/renormalization language 上。
- 未来扩展：path integrals、propagators、Wick theorem、interacting vacua、LSZ、BRST、anomalies、EFT matching 与 curved-spacetime QFT。