Linear, Tensor, and Graded Algebra

This chapter is the finite-dimensional algebra toolkit for QFT. Start with vectors, duals, and maps whenever the problem contains components or basis changes. Add tensor products for multilinear composites, a bilinear or Hermitian form for contractions and adjoints, spectral methods for modes and projectors, commutator calculus for ordering and finite transformations, and graded algebra for fermionic signs and Gaussian integrals. Not every reader needs all six pages: the right route is determined by the object being manipulated and the structure that has actually been supplied.

The chapter’s durable purpose is to replace array manipulation by typed, basis-independent statements without hiding the conventions needed to return to components. The core algebraic results are finite dimensional. The commutator page includes narrowly scoped canonical examples involving unbounded operators or smeared fields only to state the finite-matrix obstruction and mark the domain-sensitive handoff. Systematic infinite-dimensional topology, continuous spectra, unbounded-operator theory, and completed tensor products belong to later chapters, while developed field-theory applications remain in their canonical physical volumes.

Enter this chapter

There is no hard prerequisite for the chapter overview. Elementary matrix algebra is enough to begin, and a reader should repair only the capability needed for the chosen route.

Readiness check	Ready	If unsure	Repair and return
Can you distinguish a vector from its component column and a covector from an ordinary row?	Enter at vectors and duals, then follow the route for your goal.	Check whether $\alpha(v)$ remains unchanged after a non-orthogonal basis change.	Begin with Vector Spaces, Duals, Linear Maps, and Bases.
Can you explain why a bilinear map is not an ordinary linear map on $V\times W$ ?	Enter directly at tensor products.	Compare $\dim(V\oplus W)$ with $\dim(V\otimes W)$ and identify which construction linearizes bilinear maps.	Use Direct Sums, Tensor Products, and Index Structure.
Do you know which slot of $\langle v\mid w\rangle$ is conjugate-linear?	Enter at forms and adjoints.	Test how $\langle av\mid w\rangle$ changes under a complex scalar $a$ .	Review the convention bridge below, then use Bilinear and Hermitian Forms, Adjoints, and Isometries.
Can you distinguish diagonalizable, normal, and Hermitian operators?	Enter at normal forms and projectors.	Ask whether an eigenbasis must be orthonormal and whether the inner product has been specified.	Read the forms page before Normal Forms, Spectra, and Projectors.
Can you move one odd factor through another without losing the parity sign?	Enter the fermionic route.	Compute $\partial_{\theta_2}^L(\theta_1\theta_2)$ and state the sign.	Read vectors and duals, then tensor products, then Exterior and Graded Algebra, Grassmann Variables, and Berezin Integration.

For a connected repair of the first two capabilities, use Linear and tensor methods repair and return to the topic page needed for the calculation.

Choose a route

The arrows below are suggested reading order unless a hard dependency is stated explicitly.

Reader goal	Minimum coherent route	Capability at the end
First graduate encounter	vectors and duals → tensor products → forms and adjoints → spectra and projectors → commutators; add graded algebra when fermions first appear	Translate component calculations into invariant maps and name every added structure
Field multiplets and invariant couplings	vectors and duals → tensor products	Type-check sources, fields, multilinear couplings, and contractions
Mass mixing, mode decomposition, or polarizations	vectors and duals → forms and adjoints → spectra and projectors	Identify the relevant inner product, diagonalize under the correct hypothesis, and use basis-independent projectors
Finite symmetry transformations	vectors and duals → commutators and exponentials; add forms for unitary or pseudo-orthogonal preservation	Pass between a generator, its commutator action, and a finite exponential with ordering corrections visible
Fermionic Gaussian methods	vectors and duals → tensor products → graded algebra and Berezin integration	Track Koszul signs and distinguish determinant-valued from Pfaffian-valued Gaussians
Infinite-dimensional operator question	vectors and duals → forms and adjoints → finite spectra, then exit	Recognize exactly where domains, topology, and spectral measures replace matrix arguments
Conformal-bootstrap preparation	vectors and duals, then exit to convex/conic methods	Treat candidate functionals as elements of a dual space before imposing positivity or optimization structure

The chapter has five internal hard dependencies. Tensor products, forms, spectra, and commutators each require the vector-space foundation; graded algebra requires tensor products. Forms is recommended, but not hard, for the finite spectral page.

How the ideas fit together

The chapter has three layers.

Objects and constructions. Vectors, covectors, and linear maps are defined before bases. Direct sums preserve identified summands; tensor products linearize multilinear dependence.
Added geometric and dynamical structure. A bilinear or Hermitian form supplies orthogonality and an adjoint. Normality relative to a positive Hermitian form enables orthogonal spectral projectors. Commutators then control conjugation and exponential composition.
Parity and fermionic algebra. Exterior algebra turns permutation antisymmetry into multiplication. The $\mathbb Z_2$ reduction supplies the graded sign rule, and ordered top-degree coefficient extraction gives Berezin integration, determinants, and Pfaffians.

The dependencies are typed:

tensor products require the vector-space distinction between $V$ and $V^*$ ;
an adjoint depends on a chosen nondegenerate form;
the unitary spectral theorem requires a positive Hermitian form and normality;
BCH terms measure failed commutativity;
Berezin signs derive from parity and ordered odd factors; and
physical fields and states use these constructions but are developed outside this chapter.

A useful category check is to ask what changed. Choosing a basis changes only coordinates. Choosing a form adds new structure. Forming a tensor product constructs a new space. Projecting onto a symmetry type selects a subspace. Passing from finite matrices to unbounded operators changes the hypotheses and cannot be treated as a larger instance of the same calculation.

Exact chapter guide

The topic pages appear here in chapter navigation order.

Vector Spaces, Duals, Linear Maps, and Bases asks which claims survive a basis change. It establishes kernels, images, rank, dual maps, and the absence of a preferred identification $V\simeq V^*$ . Continue to tensor products for multilinear constructions or to forms when an inner product is actually present.
Direct Sums, Tensor Products, and Index Structure requires the vector-spaces-and-duals page. It explains why direct sums encode identified summands while tensor products satisfy a universal bilinear property. It supplies pure-tensor tests, contractions, and symmetric or antisymmetric projections. This page is the hard preparation for the graded-algebra topic.
Bilinear and Hermitian Forms, Adjoints, and Isometries requires the vector-spaces-and-duals page. It separates bilinear from sesquilinear forms, fixes the bra–ket linearity convention, derives Gram-matrix adjoints, and distinguishes positive from indefinite preservation. Continue to the spectral page for normal operators or to Foundations for physical Hilbert-space positivity.
Normal Forms, Spectra, and Projectors requires the vector-spaces-and-duals page; the forms page is recommended preparation. It distinguishes ordinary diagonalization, unitary spectral decomposition, Jordan form, and SVD. It makes degenerate eigenspace projectors basis independent and explains what eigenvalues omit for non-normal maps. Infinite-dimensional continuation belongs to Functional and Spectral Analysis.
Commutators and Operator Exponentials requires the vector-spaces-and-duals page. It develops the derivation rule, Hadamard lemma, BCH and Magnus expansions, and time-ordering cautions. Its canonical examples are explicitly separated from finite matrices by the trace obstruction and domain requirements.
Exterior and Graded Algebra, Grassmann Variables, and Berezin Integration requires the direct-sums-and-tensor-products page. It derives exterior and Koszul signs, fixes left derivatives and measure order, and distinguishes determinant from Pfaffian Gaussians. It ends at finite regulator; the functional-integral construction belongs to Foundations.

Convention bridge

The site-wide conventions remain in force. The recurrent local translations are:

Issue	Convention here	Invariant check
Vectors and covectors	$v\in V$ and $\alpha\in V^*$ ; index lowering requires a named form	$\alpha(v)$ is unchanged by a basis change
Complex inner products	$\langle v\mid w\rangle$ is conjugate-linear in $v$ and linear in $w$	$\langle av\mid w\rangle=a^*\langle v\mid w\rangle$
Matrix adjoints	$T^\dagger$ is defined by the forms; conjugate transpose is its matrix only in orthonormal bases	$\langle Tv\mid w\rangle=\langle v\mid T^\dagger w\rangle$
Signature	Lorentzian examples use $(+---)$	$\Lambda^{\mathsf T}\eta\Lambda=\eta$
Ordinary and graded brackets	$[A,B]=AB-BA$ ; $[a,b]_{\mathrm{gr}}=ab-(-1)^{\lvert a\rvert\lvert b\rvert}ba$	two odd elements use an anticommutator in the graded bracket
Berezin order	$d^n\theta=d\theta_n\cdots d\theta_1$ extracts the coefficient of $\theta_1\cdots\theta_n$	the normalized top-monomial integral equals $1$

One further distinction spans the chapter: a transpose belongs to the dual map and chosen bases, a Hermitian adjoint belongs to chosen forms, and a graded reversal may add parity signs. The three operations should never be identified by visual similarity.

Recurring QFT systems

A multiplet from components to modes

A finite field multiplet first appears as a vector $\phi^a e_a$ , with a source in the dual. Tensor products then type multilinear couplings. A positive kinetic form or an indefinite spacetime form determines the appropriate adjoint and contraction. A Hermitian mass matrix is resolved by orthogonal spectral projectors, while a symmetry generator acts by commutator exponentiation. The objects become richer at each step, but the recurring checks are unchanged: specify the space, specify the form, preserve the pairing, and distinguish a basis choice from a physical decomposition.

A regulated fermionic Gaussian

The fermionic route begins with tensor and exterior products, reduces degree to parity, and fixes a total order for odd generators. Berezin integration then extracts the top coefficient. With the ordered paired measure

\mathcal D(\bar\psi,\psi) = d\bar\psi_N\,d\psi_N \cdots d\bar\psi_1\,d\psi_1

fixed on the graded-algebra page, the convention is

\int\mathcal D(\bar\psi,\psi) e^{-\bar\psi A\psi} = \det A.

This system is representative of finite regulators because every series terminates. It ceases to represent the whole problem when $A$ becomes a differential operator: zero modes, boundary conditions, determinant phases, and regularization then require the relevant physics treatment.

The conformal-bootstrap handoff

The volume’s conformal-bootstrap thread uses this chapter only for linear duals: a proposed separating functional must first be a well-typed element of a dual space. Convex cones, separation theorems, semidefinite structure, and numerical certificates belong to Convex Cones, Separation, Conic Duality, and Semidefinite Programs.

Chapter synthesis

The minimum chapter-scale conclusions are:

components are representations of typed objects, so valid contractions pair compatible vector and dual factors;
direct sums and tensor products solve different universal problems;
an adjoint is meaningless until forms on the domain and codomain are specified;
normality, not diagonalizability alone, licenses an orthonormal eigenbasis and orthogonal spectral projectors;
nested commutators are the ordering data of conjugations and exponential products;
odd parity turns reordering into signs, and an ordered Berezin measure turns a quadratic Grassmann form into a determinant or Pfaffian; and
every finite-dimensional theorem stops before topology, operator domains, continuous spectrum, or continuum functional determinants enter.

Together these statements answer the organizing question: use the smallest structure that makes the desired operation intrinsic. A basis is enough to compute components; a tensor product is needed for multilinearity; a form is needed for adjoints; normality is needed for orthogonal spectral resolution; a commutator is needed for ordering; and a grading is needed for fermionic signs.

For chapter-scale reference treatments, compare Axler 2024, Chapters 3, 5, 7, and 8 on linear and spectral algebra, Conrad, n.d., Tensor Products and Bilinear Forms, PDF on the universal constructions, Hall 2015, Chapters 3 and 5 on commutators and exponentials, and Deligne and Morgan 1999, §§ 1.1 and 1.10–1.11 on graded algebra.

Review prompts

Representation change — vector and dual. Starting from $e'_a=e_bM^b{}_a$ , derive the component transformations of $v=v^ae_a$ and $\alpha=\alpha_ae^a$ . A successful response ends by showing $\alpha'_av'^a=\alpha_av^a$ . If the same matrix is assigned to both component lists, repair with the vectors-and-duals page.

Comparison — direct sum or tensor product. For two internal spaces, decide which construction describes alternative identified sectors and which describes simultaneous slots. A successful response states both dimension formulas and the tensor universal property. If the answer relies only on arrays, repair with the tensor-products page.

Derivation check — adjoint and projector. In a non-orthonormal basis with Gram matrix $G$ , reconstruct the matrix of the adjoint and explain why an orthogonal projector obeys $P^2=P=P^\dagger$ . Success requires the $G^{-1}T^{\dagger_{\mathrm{mat}}}G$ factors and a named positive form. Missing Gram matrices point back to the forms page.

Failure diagnosis — spectrum. Diagnose the claim “every diagonalizable matrix has an orthonormal eigenbasis, so its singular values are its eigenvalues.” A successful response separates diagonalizable from normal and SVD from similarity diagonalization. Repair with the spectra-and-projectors page.

Translation — finite symmetry. Given $U(\alpha)=e^{-i\alpha Q}$ , derive the infinitesimal transformation for both $U^\dagger\mathcal O U$ and $U\mathcal O U^\dagger$ . Success is the pair of opposite commutator signs. A sign obtained without specifying the conjugation order is incomplete.

Transfer — a new odd Gaussian. Given an antisymmetric $4\times4$ matrix, predict whether one real set of Grassmann variables produces a determinant or Pfaffian and state the ordering data needed to fix its sign. Success requires “Pfaffian,” the variable or measure order, and $\operatorname{Pf}(A)^2=\det A$ . Repair with the graded-algebra page.

Synthesis — finite versus field-theoretic CCR. Explain why $[q,p]=i\mathbb 1$ cannot hold for finite matrices but can be used in an infinite-dimensional representation only with domain qualifications. A successful response combines the trace check with the distinction among an algebra, a representation, and a common invariant domain.

Exit routes

For measures, convergence, Fourier transforms, and convolution, continue to Analysis, Measure, and Fourier Methods. It is a scientific continuation, not a universal prerequisite.
For matrix methods replaced by domains, completed spaces, and spectral measures, go to Functional and Spectral Analysis.
For group actions, representations, Lorentz structure, Clifford algebras, and spinors, continue to Lie Groups, Representations, and Spinors.
For canonical commutators as a physical algebra with a representation and state, use Canonical Quantization: Algebra, Representation, and State.
For physical positivity and unitary time evolution, use Hilbert Positivity and Unitary Evolution.
For regulated fermionic functional integrals, use Grassmann Functional Integrals for Free Fermions.
For capability repair before returning to this chapter, use Linear and tensor methods repair.

The Learn remediation module provides capability repair and a route back to the chapter.

References

Sheldon Axler, Linear Algebra Done Right, 4th ed., Springer, 2024, for finite-dimensional spaces, maps, duality, inner products, spectral theory, and normal forms.
Keith Conrad, Tensor Products, PDF and Bilinear Forms, PDF, University of Connecticut lecture notes, n.d., for universal tensor constructions and form-dependent adjoints.
Pierre Deligne and John W. Morgan, “Notes on Supersymmetry (following Joseph Bernstein),” in Quantum Fields and Strings: A Course for Mathematicians, Volume 1, American Mathematical Society, 1999, pp. 41–97, for graded signs and super linear algebra.
Brian C. Hall, Lie Groups, Lie Algebras, and Representations, 2nd ed., Graduate Texts in Mathematics 222, Springer, 2015, for exponential and commutator calculus.