Contribution
A reference article on Dirac’s notation. It covers kets and bras, the ket–bra identification on Hilbert space, row and column vectors, non-normalizable kets, operators acting on either side, outer products and projectors, Hermitian conjugation rules, tensor products, the resolution of the identity, and a translation into mathematicians’ notation. It adds no result.
Key insight
A bra is a functional and a ket is a vector. Writing ⟨φ|ψ⟩ as both “a bra applied to a ket” and “an inner product” is legitimate only where every continuous functional comes from a vector, which is a Hilbert space. The article says so in two places: in the identification section, with a pointer to the Riesz theorem, and in the Banach-space paragraph, where “the bracket does not have the meaning of an inner product, because the Riesz representation theorem does not apply”.
Assumptions
- Physicists’ convention. The inner product (·, ·) is antilinear in its first argument, so (φ, ·) ≡ ⟨φ| is linear.
- Complex spaces throughout, finite- or infinite-dimensional, with the separable complex Hilbert space as the setting of quantum mechanics.
Key results
The article states conventions and standard facts, not theorems:
- Identification (§Inner product and bra–ket identification). For a ket ψ and a vector φ, (φ, ψ) =: f_φ(ψ) =: ⟨φ|ψ⟩. Each ket thus gives a bra; that every bra comes from a ket is left to the cited Riesz theorem. The set of covectors ⟨φ| is called “a subspace of the dual vector space”, which on a Hilbert space is the whole continuous dual.
- Matrices. On ℂⁿ with a fixed orthonormal basis, ⟨A|B⟩ = Σ A_i* B_i, bras are rows, kets are columns, and the bra of a ket is its conjugate transpose. If A is an N × N matrix, A† is its conjugate transpose.
- Non-normalizable states. Kets of infinite norm (delta functions, plane waves) are not in the Hilbert space; the article points to the GNS construction and to rigged Hilbert spaces as broadenings.
- Outer product. (|φ⟩⟨ψ|)(x) = ⟨ψ|x⟩|φ⟩ is rank one, and for a unit ψ, |ψ⟩⟨ψ| is the orthogonal projection onto span{ψ}.
- Associativity holds for expressions in numbers, bras, kets, operators and products, but not with antilinear operators such as time reversal.
- Hermitian conjugation rules. Reverse the order and conjugate each factor: ⟨φ|ψ⟩* = ⟨ψ|φ⟩, ⟨φ|A|ψ⟩† = ⟨ψ|A†|φ⟩.
- Unit operator. For a complete orthonormal system, |ψ⟩ = Σ_i ⟨e_i|ψ⟩|e_i⟩ and Σ_i |e_i⟩⟨e_i| = 𝕀, with the continuous forms ∫dx |x⟩⟨x| = ∫dp |p⟩⟨p| = 𝕀 and ⟨x|p⟩ = e
- Mathematicians’ notation. Φ : H → H*, Φ(h) = φ_h with φ_h(g) = ⟨h, g⟩ = ⟨h|g⟩, “the embedding”.
Claims
| id | claim | strength | support |
|---|---|---|---|
| C1 | On a Hilbert space the bracket is both a functional applied to a vector and an inner product, and the identification of bras with kets rests on the Riesz theorem | moderate | stated with a pointer to the Riesz article, not proved |
| C2 | On a Banach or untopologized space bras and kets can still be written, but the bracket is not an inner product | moderate | stated, one paragraph |
| C3 | Σ_i |e_i⟩⟨e_i| is the identity for a complete orthonormal system | moderate | stated as a consequence of the basis expansion; convergence in which topology is not said |
Concepts
- Ket |v⟩. A vector of a complex vector space; physically a state.
- Bra ⟨f|. A linear form f : V → ℂ.
- Covector to |φ⟩. The bra ⟨φ| = (φ, ·) obtained from the inner product.
- Generalized eigenvector. A position or momentum “eigenket” such as |r⟩, which is not an element of the Hilbert space.
Connections
The bra–ket section of the Riesz article (LIT-230) does the same identification in the opposite convention (⟨y | x⟩ := ⟨x, y⟩, with the representer in the antilinear slot) and proves it. The article names Grassmann’s [φ|ψ] (1862) as a precursor. Its pointer to GNS (LIT-241) for non-normalizable states is discussed below.
Bearing on the record
- No THEORY, QUESTION or CLAIM cites this article. NOTE-239 names it as the notation van Rijsbergen’s book uses in its ch. 6.
- The seed’s summary needs one correction. LIT-237 says Dirac “introduced” the notation in 1939. The current lead says it was “created by Paul Dirac in his monograph, ‘The Principles of Quantum Mechanics’ published by Oxford University 1930”, while its footnote still cites Dirac (1939), “A new notation for quantum mechanics”, and a later paragraph still says the notation “was effectively established in 1939”. The 1930 sentence came in an anonymous edit of 2026-09-30 with the summary “When BRA KET notation was first created”, replacing “created by Paul Dirac in his paper, ‘A New Notation for Quantum Mechanics’ from 1939”. The record should keep 1939, from the reference the article actually gives, and not cite the lead.
- The GNS pointer is weak. The article says the definition of Hilbert space “can be broadened to accommodate” non-normalizable kets, citing the GNS construction or rigged Hilbert spaces. Rigged Hilbert spaces do that. GNS builds a Hilbert space from a state on a C*-algebra and does not house delta-function kets; the GNS article (LIT-241) says nothing about them. The record should not cite this article for a GNS treatment of improper states.
- Two internal inconsistencies, worth knowing before citing it.
- The “Pitfalls” section says |ψ⟩† = ⟨ψ| is “not correct in a technical sense”, while the “Hermitian conjugation” section uses exactly that as its first formal rule.
- In “Notation used by mathematicians”, φ_h(g) = ⟨h, g⟩ is linear in g only if the mathematicians’ inner product is linear in its second slot, but the next paragraph says mathematicians put the conjugate on the second slot (⟨φ, ψ⟩ = ∫ φ conj ψ). Under that convention the embedding should be h ↦ ⟨·, h⟩, as in LIT-230.
- No instruction for machine-learning practice; nothing for the anthology.
Limitations
- Wikipedia, under a “Cleanup reorganize” banner dated February 2023, and revised as recently as 2026-09-30. Cite it for conventions at most.
- The resolution of the identity is stated without saying in what sense the infinite sum converges (strongly, not in operator norm), and the continuous forms are formal.
- Unbounded operators (position, momentum) appear without domains.
Open questions
- None the article can settle. The rigged-Hilbert-space treatment of improper kets would need its own source if the record ever leans on it.