Today I set out to understand a deceptively simple question: why does a circuit family built from \(Z\)-rotations and nearest-neighbor \(XX\)-rotations lead so naturally to Lie algebras, Majorana operators, and the rotation group \(SO(2n)\)?
The answer is a remarkably coherent chain of ideas:
\[\begin{aligned} R^z,\;R^{xx} &\longrightarrow iZ,\;iXX \\ &\longrightarrow c_jc_k \\ &\longrightarrow \mathfrak{spin}(2n)\cong\mathfrak{so}(2n) \\ &\longrightarrow \operatorname{Spin}(2n)\longrightarrow SO(2n). \end{aligned}\]There is also a pleasant surprise at the end: inside the two-qubit matchgate group lives an ordinary \(SU(2)\), acting on the even-parity states \(\lvert 00\rangle\) and \(\lvert 11\rangle\) exactly as a single-qubit gate acts on a Bloch sphere.
This note develops that chain from the beginning. I will assume familiarity with qubits and Pauli matrices, but not with Lie theory or fermions.
Conventions
I use
\[R_q^z(\theta)=e^{+i\theta Z_q/2}, \qquad R_{q,q+1}^{xx}(\theta)=e^{+i\theta X_qX_{q+1}/2},\]and describe the action of a unitary on Majorana operators by
\[U^\dagger c_\ell U=\sum_m Q(U)_{\ell m}c_m.\]If one instead uses \(Uc_\ell U^\dagger\), every rotation below is replaced by its inverse \(Q^{-1}=Q^{\mathsf T}\). The structure is unchanged, but fixing one convention prevents sign confusion.
1. The matchgate group starts with two kinds of rotations
For \(n\) qubits, the Hilbert space is
\[\mathcal H_n=(\mathbb C^2)^{\otimes n}\cong\mathbb C^{2^n}.\]A general determinant-one unitary belongs to \(SU(2^n)\). In this note, “the matchgate group” means the connected, phase-fixed subgroup generated by
\[R_q^z(\theta)=e^{i\theta Z_q/2}\]on any qubit \(q\), and
\[R_{q,q+1}^{xx}(\theta)=e^{i\theta X_qX_{q+1}/2}\]on neighboring qubits of an open chain. Here \(\theta\) can be any real number.
Some definitions of a two-qubit matchgate also allow an arbitrary overall \(U(1)\) phase. That convention adds the central Lie-algebra direction \(iI\). I exclude it here so that the group is the spin-group image and the dimension count below is \(n(2n-1)\).
“Generated by” means taking every finite product of these gates and their inverses. The identity is obtained at \(\theta=0\), and an inverse is obtained by changing \(\theta\) to \(-\theta\). Multiplying two allowed circuits gives another allowed circuit, so these circuits form a group.
The single-qubit gate is simply
\[R^z(\theta)= \begin{pmatrix} e^{i\theta/2}&0\\ 0&e^{-i\theta/2} \end{pmatrix}.\]For the two-qubit gate, \((X\otimes X)^2=I\), so
\[R^{xx}(\theta) = \cos\frac{\theta}{2}\,I +i\sin\frac{\theta}{2}\,X\otimes X.\]It mixes \(\lvert00\rangle\) with \(\lvert11\rangle\), and separately mixes \(\lvert01\rangle\) with \(\lvert10\rangle\). It therefore preserves the parity operator
\[P=Z_1Z_2\cdots Z_n,\]whose eigenvalue is \(+1\) for computational-basis states of even Hamming weight and \(-1\) for states of odd Hamming weight. Both generators commute with \(P\), so every matchgate preserves the even- and odd-parity sectors.
For two qubits,
\[\mathcal H_{\mathrm{even}} =\operatorname{span}\{\lvert00\rangle,\lvert11\rangle\}, \qquad \mathcal H_{\mathrm{odd}} =\operatorname{span}\{\lvert01\rangle,\lvert10\rangle\}.\]That parity split is the first visible trace of the underlying fermionic structure.
2. Lie groups describe finite motion; Lie algebras describe infinitesimal motion
A Lie group is both a group and a smooth geometric space. Its elements are finite transformations that can be multiplied and inverted. A Lie algebra is the tangent space at the identity: it records the directions in which the group can initially move.
| Lie group | Lie algebra | |
|---|---|---|
| Meaning | finite transformations | infinitesimal generators |
| Operation | group multiplication | real linear combinations and commutators |
| Example | \(Q\in SO(N)\) | \(A\in\mathfrak{so}(N)\) |
| Geometry | global | local, near the identity |
| Connection | \(A\mapsto e^A\) |
Take a smooth curve of unitaries
\[U(t)=e^{tA},\qquad U(0)=I.\]Its initial velocity is \(A=U'(0)\). Expanding unitarity near \(t=0\),
\[(I+tA^\dagger)(I+tA)=I+O(t^2),\]shows that
\[A^\dagger=-A.\]Thus the Lie algebra of a unitary group consists of skew-Hermitian matrices. Physicists usually write \(A=iH\), where \(H\) is a Hermitian Hamiltonian. The two languages are equivalent:
\[\text{Hamiltonian }H \quad\longleftrightarrow\quad \text{Lie-algebra generator }iH.\]Differentiating the matchgate generators at the identity gives the initial directions
\[iZ_q, \qquad iX_qX_{q+1},\]where the harmless factors of \(1/2\) have been omitted.
Why commutators create new directions
A Lie algebra must be closed under
\[[A,B]=AB-BA.\]This is not an arbitrary rule. Four short group motions satisfy
\[e^{tA}e^{tB}e^{-tA}e^{-tB} =e^{t^2[A,B]+O(t^3)}.\]The first-order motion cancels, leaving a second-order displacement in the commutator direction. If the group can move along \(A\) and \(B\), its infinitesimal closure must therefore include \([A,B]\).
For example,
\[[iZ\otimes I,iX\otimes X]=-2iY\otimes X.\]Starting from \(iZ_1\) and \(iX_1X_2\), the Lie closure immediately produces \(iY_1X_2\). This small example will become the hidden \(SU(2)\) later.
3. Jordan–Wigner turns qubits into Majorana operators
One fermionic mode can be described by two Hermitian Majorana operators. For one qubit, the simplest example is
\[c_1=X,\qquad c_2=Y.\]They are Hermitian, square to the identity, and anticommute. In general the \(2n\) Majorana operators obey
\[\boxed{\{c_j,c_k\}=c_jc_k+c_kc_j=2\delta_{jk}I.}\]Equivalently,
\[c_j^2=I, \qquad c_jc_k=-c_kc_j\quad(j\ne k).\]There are two Majoranas per fermionic mode because a non-Hermitian annihilation operator can be decomposed as
\[a=\frac{c_1+ic_2}{2}, \qquad a^\dagger=\frac{c_1-ic_2}{2}.\]This is analogous to decomposing one complex number into two real components. Importantly, \(2n\) Majoranas do not mean \(2n\) qubits; they are \(2n\) operators acting on the same \(n\)-qubit Hilbert space.
Qubit operators on different sites normally commute, whereas fermionic operators must anticommute. The Jordan–Wigner transformation supplies the needed signs with strings of \(Z\)’s:
\[\boxed{ \begin{aligned} c_{2q-1}&=Z_1Z_2\cdots Z_{q-1}X_q,\\ c_{2q}&=Z_1Z_2\cdots Z_{q-1}Y_q. \end{aligned}}\]For two qubits this gives
\[c_1=X\otimes I,\quad c_2=Y\otimes I,\quad c_3=Z\otimes X,\quad c_4=Z\otimes Y.\]When two strings are exchanged, exactly one local \(X\)-or-\(Y\) factor crosses a \(Z\), producing the minus sign required by fermionic anticommutation.
Why Majorana bilinears belong to a unitary Lie algebra
For \(j\ne k\),
\[(c_jc_k)^\dagger=c_kc_j=-c_jc_k,\]so \(c_jc_k\) is skew-Hermitian. It also satisfies
\[(c_jc_k)^2=-I,\]and hence
\[e^{\alpha c_jc_k} =\cos\alpha\,I+\sin\alpha\,c_jc_k.\]The Hermitian Hamiltonian version is \(ic_jc_k\). This is exactly the algebraic form of a quadratic free-fermion Hamiltonian.
4. The original gates are adjacent Majorana couplings
The Jordan–Wigner definitions make the two matchgate generators especially simple. On a single qubit,
\[c_{2q-1}c_{2q}=iZ_q,\]so
\[R_q^z(\theta) =e^{\frac{\theta}{2}c_{2q-1}c_{2q}}.\]Across neighboring qubits,
\[c_{2q}c_{2q+1}=iX_qX_{q+1},\]so
\[R_{q,q+1}^{xx}(\theta) =e^{\frac{\theta}{2}c_{2q}c_{2q+1}}.\]The local \(Z\)-rotations provide the pairs
\[c_1c_2,\;c_3c_4,\;c_5c_6,\ldots,\]and the neighboring \(XX\)-rotations provide
\[c_2c_3,\;c_4c_5,\;c_6c_7,\ldots.\]Together they give every edge of a path of \(2n\) Majoranas:
\[1-2-3-4-\cdots-2n.\]The key commutator identity for three distinct indices is
\[\boxed{[c_ac_b,c_bc_c]=2c_ac_c.}\]Indeed, the product in one order is \(c_ac_c\), while the product in the opposite order is \(-c_ac_c\). Thus the two edges \(a-b\) and \(b-c\) generate the shortcut \(a-c\).
Repeated commutators extend shortcuts across the entire path. Starting from adjacent couplings, we can generate every \(c_jc_k\) with \(j<k\). The full matchgate Lie algebra is therefore
\[\boxed{ \mathfrak m_n =\operatorname{span}_{\mathbb R} \{c_jc_k:1\le j<k\le2n\}. }\]The coefficients are real because a real linear combination of skew-Hermitian generators remains skew-Hermitian. There is one independent generator for every unordered pair of Majorana indices, so
\[\boxed{ \dim\mathfrak m_n =\binom{2n}{2} =n(2n-1). }\]For comparison:
| Qubits \(n\) | Majoranas \(2n\) | \(\dim\mathfrak m_n\) | \(\dim\mathfrak{su}(2^n)\) |
|---|---|---|---|
| 1 | 2 | 1 | 3 |
| 2 | 4 | 6 | 15 |
| 3 | 6 | 15 | 63 |
| 4 | 8 | 28 | 255 |
The matchgate algebra grows as \(O(n^2)\), while the full \(n\)-qubit algebra grows as \(4^n-1\). This dimension gap is the first sign that matchgates admit a polynomial-size description.
5. The same algebra appears as real infinitesimal rotations
The special orthogonal group is
\[SO(N)=\{Q\in\mathbb R^{N\times N}:Q^{\mathsf T}Q=I,\ \det Q=1\}.\]Its elements are finite, orientation-preserving real rotations. To find its Lie algebra, take a curve \(Q(t)\in SO(N)\) with \(Q(0)=I\), differentiate
\[Q(t)^{\mathsf T}Q(t)=I,\]and set \(A=Q'(0)\). At \(t=0\),
\[A^{\mathsf T}+A=0.\]Therefore
\[\mathfrak{so}(N) =\{A\in\mathbb R^{N\times N}:A^{\mathsf T}=-A\}.\]For \(N=2n\), a real antisymmetric matrix has zero diagonal, and each entry below the diagonal is fixed by the corresponding entry above it. The free parameters are again the unordered pairs \(j<k\):
\[\dim\mathfrak{so}(2n)=\binom{2n}{2}=n(2n-1).\]For every \(j<k\), define
\[\boxed{L_{jk}=e_je_k^{\mathsf T}-e_ke_j^{\mathsf T},}\]or entrywise,
\[(L_{jk})_{\ell m} =\delta_{j\ell}\delta_{km} -\delta_{jm}\delta_{k\ell}.\]The matrix \(L_{jk}\) has a \(+1\) in position \((j,k)\), a \(-1\) in position \((k,j)\), and zeros elsewhere. The collection \(\{L_{jk}:j<k\}\) is a basis of \(\mathfrak{so}(2n)\).
On the \(j\)-\(k\) coordinate plane,
\[L_{jk}\;\widehat{=}\; \begin{pmatrix}0&1\\-1&0\end{pmatrix},\]so exponentiation produces the ordinary plane rotation
\[e^{\theta L_{jk}}\;\widehat{=}\; \begin{pmatrix} \cos\theta&\sin\theta\\ -\sin\theta&\cos\theta \end{pmatrix}.\]The matchgate and rotation algebras are related by
\[\boxed{\phi(c_jc_k)=2L_{jk}.}\]This map is linear, bijective, and preserves commutators. For example,
\[[c_1c_2,c_2c_3]=2c_1c_3, \qquad [L_{12},L_{23}]=L_{13},\]and both sides agree after applying \(\phi\). Consequently,
\[\boxed{\mathfrak{spin}(2n)\cong\mathfrak{so}(2n).}\]The matrices on the two sides have very different sizes, but their linear and commutator structures are identical.
6. A matchgate rotates the Majorana operators
Let
\[U=e^A, \qquad A=\sum_{j<k}\alpha_{jk}c_jc_k, \qquad \alpha_{jk}\in\mathbb R.\]The essential commutator with one Majorana is
\[\boxed{ [c_jc_k,c_\ell] =2\delta_{k\ell}c_j-2\delta_{j\ell}c_k. }\]It is always a real linear combination of individual Majoranas. Therefore conjugation by \(U\) never leaves the \(2n\)-dimensional real operator space
\[V_{\mathrm{Maj}} =\operatorname{span}_{\mathbb R}\{c_1,\ldots,c_{2n}\}.\]There must be a real \(2n\times2n\) matrix \(Q(U)\) such that
\[\boxed{ U^\dagger c_\ell U =\sum_{m=1}^{2n}Q(U)_{\ell m}c_m. }\]Write \(\Pi(U)=Q(U)\) for the resulting map from spin transformations to real rotations.
For the elementary gate
\[U_{jk}(\theta)=e^{\theta c_jc_k/2},\]the differential equations close on \(c_j\) and \(c_k\), giving
\[\begin{aligned} U_{jk}(\theta)^\dagger c_jU_{jk}(\theta) &=\cos\theta\,c_j+\sin\theta\,c_k,\\ U_{jk}(\theta)^\dagger c_kU_{jk}(\theta) &=-\sin\theta\,c_j+\cos\theta\,c_k. \end{aligned}\]All other Majoranas remain fixed. Hence
\[e^{\theta c_jc_k/2} \quad\longleftrightarrow\quad e^{\theta L_{jk}}.\]This also explains the factor \(2\) in \(\phi(c_jc_k)=2L_{jk}\).
Why is \(Q\) real and special orthogonal?
-
The coefficients can be extracted as
\[Q_{\ell m}=2^{-n}\operatorname{Tr}\!\left(c_mU^\dagger c_\ell U\right),\]which is real because both factors inside the trace are Hermitian.
- Unitary conjugation preserves the Majorana anticommutation relations. Substitution gives \(QQ^{\mathsf T}=I\), so \(Q\in O(2n)\).
- Matchgates are connected continuously to the identity. Since an orthogonal determinant can only be \(+1\) or \(-1\), it cannot jump away from \(+1\) along that path. Thus \(Q\in SO(2n)\).
With the chosen convention, multiplication is respected:
\[\Pi(UV)=\Pi(U)\Pi(V).\]More generally,
\[U=\exp\!\left(\sum_{j<k}\alpha_{jk}c_jc_k\right) \quad\longmapsto\quad Q=\exp\!\left(\sum_{j<k}2\alpha_{jk}L_{jk}\right).\]A \(2^n\times2^n\) complex unitary has been replaced by a \(2n\times2n\) real rotation that determines its Majorana action.
7. Why \(\operatorname{Spin}(2n)\) is a double cover of \(SO(2n)\)
The Lie algebras match exactly, but the Lie groups have different global topology. The simplest way to see this is to rotate one Majorana plane.
For the real rotation,
\[e^{(\theta+2\pi)L_{jk}}=e^{\theta L_{jk}}.\]On the spin side, since \((c_jc_k)^2=-I\),
\[\begin{aligned} e^{(\theta+2\pi)c_jc_k/2} &=e^{\theta c_jc_k/2}e^{\pi c_jc_k}\\ &=-e^{\theta c_jc_k/2}. \end{aligned}\]Thus a \(2\pi\) rotation returns \(Q\) to itself but changes \(U\) to \(-U\). Only after \(4\pi\) does the spin element return exactly to its starting point.
The sign disappears under conjugation:
\[(-U)^\dagger c_\ell(-U)=U^\dagger c_\ell U.\]In fact, the kernel of the map \(\Pi:U\mapsto Q(U)\) is precisely
\[\ker\Pi=\{I,-I\}.\]If \(Q(U)=I\), then \(U\) commutes with every Majorana. The Majoranas generate the full Pauli operator algebra, so \(U\) must be a scalar; inside the spin group the possibilities are \(I\) and \(-I\). Therefore
\[\boxed{ \operatorname{Spin}(2n)/\{\pm I\}\cong SO(2n). }\]This is what “double cover” means: every real rotation \(Q\) has two lifts, \(U\) and \(-U\). Locally, near the identity, the correspondence is one-to-one, which is why the Lie algebras are isomorphic. Globally, the sign ambiguity remains, which is why the groups are not.
Usually the two lifts differ only by an unobservable global phase. The distinction can become physical if \(U\) is inserted into a controlled-\(U\), because the sign is then a relative phase between control branches.
8. The hidden \(SU(2)\) inside two-qubit matchgates
The ordinary single-qubit Lie algebra is
\[\mathfrak{su}(2)=\operatorname{span}_{\mathbb R}\{iX,iY,iZ\},\]with commutators
\[[iX,iY]=-2iZ, \qquad [iY,iZ]=-2iX, \qquad [iZ,iX]=-2iY.\]Now consider the two-qubit operators
\[iX_1X_2, \qquad iY_1X_2, \qquad iZ_1.\]They satisfy exactly the same commutation table. The map
\[\boxed{ \begin{aligned} iX&\longmapsto iX_1X_2,\\ iY&\longmapsto iY_1X_2,\\ iZ&\longmapsto iZ_1 \end{aligned}}\]is therefore a Lie-algebra isomorphism onto a three-dimensional subalgebra of the two-qubit matchgate algebra. Because \(SU(2)\) is simply connected, this infinitesimal map integrates consistently to a Lie-group representation:
\[e^{i\theta X/2}\longmapsto R_{1,2}^{xx}(\theta), \qquad e^{i\theta Z/2}\longmapsto R_1^z(\theta).\]The even-parity sector is a logical qubit
Define
\[\lvert0_L\rangle=\lvert00\rangle, \qquad \lvert1_L\rangle=\lvert11\rangle.\]Restricted to this subspace,
\[X_1X_2=X_L, \qquad Y_1X_2=Y_L, \qquad Z_1=Z_L.\]In the logical basis \(\{\lvert00\rangle,\lvert11\rangle\}\),
\[R^z_{1,\mathrm{even}}(\theta) = \begin{pmatrix} e^{i\theta/2}&0\\ 0&e^{-i\theta/2} \end{pmatrix},\]and
\[R^{xx}_{1,2,\mathrm{even}}(\theta) = \begin{pmatrix} \cos(\theta/2)&i\sin(\theta/2)\\ i\sin(\theta/2)&\cos(\theta/2) \end{pmatrix}.\]These are exactly \(e^{i\theta Z_L/2}\) and \(e^{i\theta X_L/2}\). The two-qubit matchgate subgroup therefore acts on the even-parity sector as an ordinary Bloch-sphere \(SU(2)\).
For this embedded subgroup, the same action occurs on the odd-parity sector. In the parity-ordered basis
\[\{\lvert00\rangle,\lvert11\rangle,\lvert01\rangle,\lvert10\rangle\},\]the representation is \(V\oplus V\): two copies of the standard two-dimensional \(SU(2)\) representation.
There is a second, overlapping copy obtained from \(Z_2\):
\[iX\mapsto iX_1X_2, \qquad iY\mapsto iX_1Y_2, \qquad iZ\mapsto iZ_2.\]The two copies share the \(XX\)-rotation. Together, their Lie closure is the full six-dimensional two-qubit matchgate algebra
\[\mathfrak m_2 =\operatorname{span}_{\mathbb R} i\{Z_1,Z_2,XX,YY,XY,YX\}.\]9. Why the hidden \(SU(2)\) is useful for synthesis
The embedded \(SU(2)\) is not only a geometric curiosity. It lets us translate ordinary one-qubit synthesis identities directly into matchgate identities.
For example, the determinant-one \(T\)-type rotation maps as
\[e^{i\pi Z/8} \longmapsto e^{i\pi Z_1/8} =R_1^z\!\left(\frac{\pi}{4}\right).\]Let
\[W=\frac{Y+Z}{\sqrt2}.\]Since \(W^2=I\), \(e^{i\pi W/2}=iW\). Under the same representation,
\[iW \longmapsto i\frac{Y_1X_2+Z_1}{\sqrt2}.\]Using
\[Y_1X_2+Z_1=(I+iX_1X_2)Z_1,\]we obtain the matchgate factorization
\[\boxed{ i\frac{Y_1X_2+Z_1}{\sqrt2} =R_{1,2}^{xx}\!\left(\frac{\pi}{2}\right)R_1^z(\pi). }\]The order shown matters: reversing the two factors changes the sign of the \(Y_1X_2\) term.
So methods for approximating arbitrary single-qubit rotations can be reused inside each logical matchgate \(SU(2)\). Applying the two overlapping copies along a chain reaches the full matchgate group.
One sign warning is worth making explicit. With the convention \(R^z(\theta)=e^{+i\theta Z/2}\) and the conventional \(T=\operatorname{diag}(1,e^{i\pi/4})\),
\[R^z\!\left(\frac{\pi}{4}\right)=e^{i\pi/8}T^\dagger, \qquad e^{-i\pi/8}T=R^z\!\left(-\frac{\pi}{4}\right).\]Both \(T\) and \(T^\dagger=T^7\) generate the same discrete subgroup, but writing the rotation angle explicitly avoids mixing conventions.
10. The computational payoff
The final picture separates three levels that are easy to conflate:
\[\begin{array}{ccc} \text{quadratic Hamiltonians} &\xrightarrow{\times i}& \mathfrak{spin}(2n)\\[1mm] &&\downarrow\exp\\[1mm] &&\operatorname{Spin}(2n) \xrightarrow{\text{Majorana conjugation}}SO(2n). \end{array}\]- The Hamiltonians are Hermitian quadratic forms \(i\sum_{j<k}h_{jk}c_jc_k\).
- The Lie algebra is the real span of the skew-Hermitian bilinears \(c_jc_k\).
- The matchgate group contains the finite unitaries obtained by exponentiation and multiplication.
- The real rotation \(Q\in SO(2n)\) records how a matchgate transforms the Majorana operators.
- The map from spin transformations to real rotations forgets only the global sign \(U\leftrightarrow -U\).
This is why an exponentially large \(2^n\times2^n\) matchgate unitary can be handled through a \(2n\times2n\) real matrix with only \(n(2n-1)\) independent parameters. The simplification is not a numerical accident. It is the concrete computational consequence of the Lie-algebra isomorphism
\[\mathfrak{spin}(2n)\cong\mathfrak{so}(2n)\]and the double-cover map
\[\operatorname{Spin}(2n)\longrightarrow SO(2n).\]What I want to remember
- A matchgate is not one special gate; it is any circuit generated by continuous \(Z\)- and nearest-neighbor \(XX\)-rotations.
- A Lie group contains finite transformations. Its Lie algebra contains infinitesimal generators and is closed under real linear combinations and commutators.
- Jordan–Wigner maps \(n\) qubits to \(2n\) anticommuting Majorana operators.
- The original gates generate adjacent Majorana bilinears, and commutators generate every \(c_jc_k\).
- There are \(\binom{2n}{2}=n(2n-1)\) such directions, exactly matching \(\dim\mathfrak{so}(2n)\).
- Conjugating Majoranas turns a matchgate into a real rotation \(Q\in SO(2n)\).
- The Lie algebras are isomorphic, but the groups differ globally: \(U\) and \(-U\) give the same \(Q\).
- On \(\operatorname{span}\{\lvert00\rangle,\lvert11\rangle\}\), a two-qubit matchgate subgroup behaves exactly like a single logical qubit.
References and further reading
- B. Casas, P. Braccia, É. Gouzien, M. Cerezo, and D. García-Martín, “Matchgate synthesis via Clifford matchgates and \(T\) gates”, arXiv:2602.05425 (2026). This was the motivation for the present learning note.
- R. Jozsa and A. Miyake, “Matchgates and classical simulation of quantum circuits”, Proceedings of the Royal Society A 464, 3089–3106 (2008).
- E. Knill, “Fermionic Linear Optics and Matchgates”, arXiv:quant-ph/0108033 (2001).
- B. M. Terhal and D. P. DiVincenzo, “Classical simulation of noninteracting-fermion quantum circuits”, Physical Review A 65, 032325 (2002).
- P. Jordan and E. Wigner, “Über das Paulische Äquivalenzverbot”, Zeitschrift für Physik 47, 631–651 (1928).