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Antisymmetric tensor

An antisymmetric tensor, also known as an alternating tensor, is a mathematical object that generalizes scalars, vectors, and matrices by transforming under coordinate changes in a specific way, with the defining property that it changes sign upon interchanging any two of its indices—for instance, for a rank-2 tensor T_{ij}, T_{ij} = -T_{ji}.^[1] This antisymmetry implies that the diagonal components vanish, and for a rank-2 antisymmetric tensor in three dimensions, there are exactly three independent components, allowing it to be dual to a pseudovector via the Levi-Civita symbol, such that T_{ij} = \epsilon_{ijk} B_k for some vector B_k.^[1]^[2] In higher dimensions or for higher-rank tensors, antisymmetry can be partial (with respect to specific index pairs) or total (with respect to all pairs), limiting the number of independent components to the binomial coefficient \binom{n}{p} for a totally antisymmetric rank-p tensor in n-dimensional space, where p \leq n.^[1] For example, a rank-2 tensor can be uniquely decomposed into its symmetric part (which includes a trace) and antisymmetric part, facilitating analysis in applications like continuum mechanics and representation theory.^[1] The totally antisymmetric Levi-Civita tensor \epsilon_{i_1 \dots i_n}, which is +1 for even permutations, -1 for odd, and 0 otherwise of an ordered index set, serves as a fundamental example and is invariant under proper rotations but changes sign under reflections, classifying it as a pseudotensor.^[3] Antisymmetric tensors play a central role in physics and geometry, representing quantities like the electromagnetic field strength tensor F_{\mu\nu} in relativity, which encodes electric and magnetic fields and satisfies F_{\mu\nu} = -F_{\nu\mu}, or the cross product in three dimensions, where \mathbf{a} \times \mathbf{b} corresponds to the antisymmetric tensor with components a_i b_j - a_j b_i.^[4] In differential geometry, they underpin exterior algebra and differential forms, essential for integration over manifolds and Stokes' theorem.^[1] Their properties extend to quantum mechanics and string theory, where higher-rank antisymmetric tensors describe fermionic states or gauge fields.^[5]

Basic Concepts

Definition

Tensors can be viewed as multilinear maps from powers of a vector space and its dual to the scalar field.^[6] A tensor T of type (0,k) over a vector space V is antisymmetric (or alternating) if it is a multilinear map T: V^k \to \mathbb{F} (where \mathbb{F} is the scalar field) satisfying T(v_1, \dots, v_i, \dots, v_j, \dots, v_k) = -T(v_1, \dots, v_j, \dots, v_i, \dots, v_k) for all i < j and all vectors v_1, \dots, v_k \in V.^[6] This pairwise interchange condition implies the full antisymmetry property: for any permutation \sigma of \{1, \dots, k\}, T(v_{\sigma(1)}, \dots, v_{\sigma(k)}) = \operatorname{sgn}(\sigma) T(v_1, \dots, v_k), where \operatorname{sgn}(\sigma) is the sign of the permutation \sigma. Such tensors generate the exterior power \bigwedge^k V^*.^[6] For mixed tensors of type (k,l), antisymmetry is defined with respect to specific pairs of indices, whether contravariant or covariant, such that interchanging those indices results in a sign change. In contrast, symmetric tensors satisfy the corresponding condition with a plus sign for all permutations.^[6]

Relation to Symmetric Tensors

A symmetric tensor is a multilinear map T: V^k \to \mathbb{F} that remains unchanged upon interchanging any two arguments: T(v_1, \dots, v_i, \dots, v_j, \dots, v_k) = T(v_1, \dots, v_j, \dots, v_i, \dots, v_k) for all v_\ell \in V and i < j.^[7] In the vector space of all k-covariant tensors on an n-dimensional vector space V, the subspace of totally antisymmetric tensors has dimension \binom{n}{k}, corresponding to the number of independent components. In contrast, the subspace of totally symmetric tensors has dimension \binom{n + k - 1}{k}.^[7]^[8] When the space of k-tensors is endowed with an inner product induced from an inner product on V (such as the extension of the Frobenius inner product for components), the symmetric and totally antisymmetric subspaces are orthogonal, as the inner product of a symmetric tensor S and a totally antisymmetric tensor A vanishes due to opposing symmetry under index permutations. For rank-2 tensors, any T can be uniquely decomposed as T = S + A, where S is the symmetrized part and A the antisymmetrized part. For higher ranks, more general decompositions into irreducible representations are required.^[9]

Notation and Conventions

Index Notation

In index notation, an antisymmetric tensor of rank k in a vector space of dimension n is expressed as T^{i_1 \dots i_k}, where the components satisfy the antisymmetry condition T^{i_1 \dots i_m \dots i_p \dots i_k} = -T^{i_1 \dots i_p \dots i_m \dots i_k} upon interchanging any pair of indices i_m and i_p with m < p.^[10] This property implies that the tensor changes sign under odd permutations of its indices, and the Einstein summation convention is employed, whereby repeated indices (one upper and one lower) imply summation over their range from 1 to n.^[11] In contrast, a symmetric tensor obeys T^{ij} = T^{ji}.^[12] For fully antisymmetric tensors, which are antisymmetric under interchange of any pair of indices, the Levi-Civita symbol \varepsilon_{i_1 \dots i_n} provides a fundamental characterization in n dimensions; this pseudotensor is defined such that \varepsilon_{i_1 \dots i_n} = +1 for even permutations of $1,2,\dots,n, -1 for odd permutations, and 0 if any indices repeat.^[13] The Levi-Civita symbol itself is a fully antisymmetric object, and it is used to construct or express other fully antisymmetric tensors through contractions. As an illustrative form, the components of a fully antisymmetric tensor A_{i_1 \dots i_k} can be related to another tensor B^{j_1 \dots j_{n-k}} via the Levi-Civita symbol as

A_{i_1 \dots i_k} = \frac{1}{k!(n-k)!} \varepsilon_{i_1 \dots i_k j_1 \dots j_{n-k}} B^{j_1 \dots j_{n-k}},

where the factor accounts for the antisymmetrization over the respective index sets. In the context of Riemannian manifolds, conventions for upper and lower indices on antisymmetric tensors follow the metric tensor g_{ij}, which raises contravariant indices (e.g., T^{i_1 \dots i_k} = g^{i_1 j_1} \cdots T_{j_1 \dots j_k}) and lowers covariant indices (e.g., T_{i_1 \dots i_k} = g_{i_1 j_1} \cdots T^{j_1 \dots j_k}); the Levi-Civita tensor is then defined with lower indices as \varepsilon_{i_1 \dots i_n} = \sqrt{|g|} \, \tilde{\varepsilon}_{i_1 \dots i_n}, where \tilde{\varepsilon} is the symbol and g = \det(g_{ij}), ensuring compatibility with the manifold's geometry.^[14] The upper-index version is \varepsilon^{i_1 \dots i_n} = \frac{1}{\sqrt{|g|}} \, \tilde{\varepsilon}^{i_1 \dots i_n}.^[14]

Multilinear Map Notation

In multilinear algebra, an antisymmetric tensor of rank k on a vector space V over a field of characteristic not equal to 2 is represented as an alternating multilinear map T: V^k \to \mathbb{F}, where \mathbb{F} is the base field, belonging to the k-th exterior power of the dual space \wedge^k V^*.^[15] This coordinate-free perspective emphasizes the tensor's role in the exterior algebra, where \wedge^k V^* consists precisely of such maps that vanish whenever any two arguments are identical and change sign under transposition of adjacent arguments.^[16] The defining alternating property is that for any permutation \sigma \in S_k, the symmetric group on k elements,

T(\sigma \mathbf{v}_1, \dots, \sigma \mathbf{v}_k) = \operatorname{sgn}(\sigma) \, T(\mathbf{v}_1, \dots, \mathbf{v}_k),

where \mathbf{v}_i \in V and \operatorname{sgn}(\sigma) is the sign of the permutation.^[15] This total antisymmetry ensures the map is zero on repeated inputs, distinguishing it from partially antisymmetric tensors, which are antisymmetric only with respect to specific pairs or subsets of arguments but may not vanish under all repetitions.^[17] To obtain an antisymmetric tensor from a general multilinear map T: V^k \to \mathbb{F}, one applies the alternation operator, a projection onto \wedge^k V^* defined by

\operatorname{Alt}(T)(\mathbf{v}_1, \dots, \mathbf{v}_k) = \frac{1}{k!} \sum_{\sigma \in S_k} \operatorname{sgn}(\sigma) \, T(\mathbf{v}_{\sigma(1)}, \dots, \mathbf{v}_{\sigma(k)}).

This operator is idempotent, meaning \operatorname{Alt}^2 = \operatorname{Alt}, and yields the unique antisymmetric part of T.^[17] In index notation, the components of \operatorname{Alt}(T) correspond to the fully antisymmetrized expression over all indices.^[16]

Properties

Algebraic Properties

Antisymmetric tensors, also known as alternating tensors, exhibit several key algebraic properties arising from their defining antisymmetry condition, T(v_1, \dots, v_i, \dots, v_j, \dots, v_k) = -T(v_1, \dots, v_j, \dots, v_i, \dots, v_k) for any pair of indices i < j.^[18] A fundamental property is that antisymmetric tensors vanish upon contraction in ways that reflect their antisymmetry. For a rank-two antisymmetric tensor A_{ij}, the diagonal components satisfy A_{ii} = -A_{ii}, implying A_{ii} = 0 for all i, and thus the trace \operatorname{tr}(A) = \sum_i A_{ii} = 0.^[12] More generally, the contraction with a vector v^j yields a covector w_i = A_{ij} v^j, and the associated bilinear form v^i A_{ij} v^j = 0 for any vector v, since swapping the indices changes the sign but leaves the expression unchanged.^[12] For higher-rank antisymmetric tensors, viewed as alternating multilinear maps T: V^k \to \mathbb{R} on a vector space V, the map vanishes whenever any two arguments are identical: T(v, v, w_1, \dots, w_{k-2}) = 0 for any vectors v, w_1, \dots, w_{k-2} \in V.^[18] This follows directly from antisymmetry, as T(v, v, \dots) = -T(v, v, \dots), forcing the value to be zero; it suffices to check adjacent repeated arguments due to multilinearity.^[18] Under the action of the general linear group \mathrm{GL}(n), a rank-k antisymmetric tensor transforms according to the k-th alternating (or exterior) representation \wedge^k \mathrm{GL}(n), which is the k-th exterior power of the standard representation on \mathbb{R}^n.^[19] This representation is irreducible for k \leq n and acts via the k \times k minors of the transformation matrix. The algebra of antisymmetric tensors is equipped with the wedge product \wedge, which serves as the tensor product in the exterior algebra. For p-forms \alpha and q-forms \beta, the wedge product is defined by its action on vectors as

(\alpha \wedge \beta)(v_1, \dots, v_{p+q}) = \frac{1}{p! \, q!} \sum_{\sigma \in S_{p+q}} \operatorname{sgn}(\sigma) \, \alpha(v_{\sigma(1)}, \dots, v_{\sigma(p)}) \, \beta(v_{\sigma(p+1)}, \dots, v_{\sigma(p+q)}),

where the sum is over all permutations \sigma of \{1, \dots, p+q\}, and \operatorname{sgn}(\sigma) is the sign of the permutation.^[18] This operation is bilinear, associative, and antisymmetric, ensuring the result remains an antisymmetric tensor of rank p+q.^[18]

Determinant and Pfaffian Relations

In multilinear algebra, the determinant of an n \times n matrix A with column vectors v_1, \dots, v_n is given by the evaluation of a fully antisymmetric n-tensor, known as the volume form, on these vectors, which yields the signed oriented volume of the parallelepiped they span.^[20] Specifically, for a fully antisymmetric tensor \omega of rank n in an n-dimensional vector space, it defines an oriented volume via

\omega(v_1, \dots, v_n) = \det(v_1, \dots, v_n),

where the determinant provides a scalar measure invariant under basis changes, up to the sign determined by the orientation.^[20] The space of fully antisymmetric n-tensors in n dimensions is one-dimensional up to scalar multiples, as the exterior power \Lambda^n V for a vector space V of dimension n has dimension \binom{n}{n} = 1, ensuring that any such tensor is proportional to a fixed volume form.^[10] For even rank, antisymmetric tensors correspond to skew-symmetric matrices, where the Pfaffian provides a square root of the determinant. For a $2n \times 2n skew-symmetric matrix K, the Pfaffian satisfies \mathrm{Pf}(K)^2 = \det(K), and is defined by

\mathrm{Pf}(K) = \frac{1}{2^n n!} \sum_{\sigma \in S_{2n}} \mathrm{sgn}(\sigma) \prod_{i=1}^n K_{\sigma(2i-1), \sigma(2i)},

where the sum runs over all permutations \sigma of \{1, \dots, 2n\}, and \mathrm{sgn}(\sigma) is the sign of the permutation.^[21] This relation highlights the Pfaffian as an antisymmetric analogue to the determinant, useful in contexts requiring oriented volumes for even-dimensional skew forms.^[21]

Decompositions

Symmetric-Antisymmetric Decomposition

Any tensor can be decomposed into its symmetric and antisymmetric components using projection operators derived from the action of the symmetric group on the indices. For a rank-two covariant tensor T_{ij}, the symmetric part is given by

S_{ij} = \frac{1}{2} (T_{ij} + T_{ji}),

and the antisymmetric part by

A_{ij} = \frac{1}{2} (T_{ij} - T_{ji}),

such that T_{ij} = S_{ij} + A_{ij}.^[1] This decomposition is unique and spans the entire space of rank-two tensors as a direct sum of the symmetric and antisymmetric subspaces. For higher-rank tensors of order k, the generalization employs the symmetrizer and antisymmetrizer projectors. The symmetrizer projects onto the totally symmetric subspace:

\text{Sym}(T)_{i_1 \dots i_k} = \frac{1}{k!} \sum_{\sigma \in S_k} T_{i_{\sigma(1)} \dots i_{\sigma(k)}},

where S_k is the symmetric group on k elements, and the sum averages over all permutations \sigma of the indices. The antisymmetrizer projects onto the totally antisymmetric subspace:

\text{Alt}(T)_{i_1 \dots i_k} = \frac{1}{k!} \sum_{\sigma \in S_k} \operatorname{sign}(\sigma) \, T_{i_{\sigma(1)} \dots i_{\sigma(k)}},

with \operatorname{sign}(\sigma) = \pm 1 depending on the parity of the permutation. For k=2, these reduce to the rank-two formulas above. In general, any tensor admits a partial decomposition T = \text{Sym}(T) + \text{Alt}(T) + R, where R captures components with mixed symmetries; the full tensor space decomposes as a direct sum of the totally symmetric subspace, the totally antisymmetric subspace, and subspaces of other symmetry types corresponding to irreducible representations of S_k.^[22] These subspaces are orthogonal with respect to the Frobenius inner product on the tensor space, defined by

\langle T, U \rangle = \sum_{i_1, \dots, i_k} T^{i_1 \dots i_k} U_{i_1 \dots i_k}.

To see this, note that the inner product is invariant under simultaneous permutation of indices in T and U. The pairing \langle \text{Sym}(T), \text{Alt}(U) \rangle then averages the invariant inner product over all permutations with the sign character of the antisymmetric representation, yielding zero due to the orthogonality of irreducible characters of S_k under the standard inner product on the group algebra. This ensures the projectors are orthogonal, guaranteeing uniqueness in the decomposition.^[22]

Cartan Decomposition

In the context of Lie algebras associated with orthogonal and symplectic groups, the Cartan decomposition provides a fundamental splitting that highlights the role of antisymmetric tensors as generators. For the special orthogonal Lie algebra \mathfrak{so}(p,q), which generalizes \mathfrak{so}(n) = \mathfrak{so}(n,0), the elements are real (p+q) \times (p+q) matrices X satisfying X^T I_{p,q} + I_{p,q} X = 0, where I_{p,q} = \diag(I_p, -I_q); this condition encodes antisymmetry with respect to the indefinite bilinear form defined by I_{p,q}. The Cartan involution is given by \theta(X) = -I_{p,q} X^T I_{p,q}, an involutive automorphism (\theta^2 = \id) that preserves the Lie algebra structure. The +1-eigenspace is k = \{X \in \mathfrak{so}(p,q) \mid \theta(X) = X\}, the Lie algebra of the maximal compact subgroup \SO(p) \times \SO(q), comprising block-diagonal components with skew-symmetric blocks in the p- and q-sectors. The -1-eigenspace is p = \{X \in \mathfrak{so}(p,q) \mid \theta(X) = -X\}, consisting of block off-diagonal matrices where the off-diagonal blocks are symmetric (up to sign conventions in the basis). This yields the direct sum decomposition \mathfrak{so}(p,q) = k \oplus p, with [k, p] \subseteq p and [p, p] \subseteq k.^[23]^[24] The decomposition is orthogonal with respect to the Killing form B(X,Y) = \tr(\ad_X \ad_Y), the unique (up to scalar) invariant nondegenerate symmetric bilinear form on the semisimple Lie algebra, satisfying B(k, p) = 0, with B negative definite on k and positive definite on p. This orthogonality follows from the ad-invariance of B and the action of \theta, since B(\theta(X), \theta(Y)) = B(X,Y) implies the eigenspaces are mutually orthogonal. For the compact real form \mathfrak{so}(n), the involution simplifies to \theta(X) = -X^T = X (as X^T = -X), yielding the trivial decomposition with p = \{0\} and k = \mathfrak{so}(n), underscoring the purely "antisymmetric" nature of the generators. The Cartan subalgebra \mathfrak{a} (maximal abelian subalgebra of diagonalizable elements) is typically chosen within p for noncompact forms, facilitating root space decompositions.^[23]^[24] A parallel structure holds for the symplectic Lie algebra \mathfrak{sp}(2n, \mathbb{R}), consisting of real $2n \times 2n matrices X satisfying X^T J + J X = 0, where J = \begin{pmatrix} 0 & I_n \\ -I_n & 0 \end{pmatrix} defines the antisymmetric symplectic form; elements can be viewed as "antisymmetric" with respect to this form. The Cartan involution is \theta(X) = -J X^T J, with +1-eigenspace k \cong \mathfrak{u}(n) (the Lie algebra of the maximal compact \U(n)) and -1-eigenspace p comprising purely imaginary symmetric matrices in a suitable block basis. The decomposition \mathfrak{sp}(2n, \mathbb{R}) = k \oplus p is again orthogonal under the Killing form B, which is negative definite on k and positive definite on p. There are two Cartan decomposition types for \mathfrak{sp}(n) (CI and CII), corresponding to different real forms and involutions, but the orthogonality property persists.^[25]^[23] This framework extends to higher-rank antisymmetric tensors in irreducible representations of \O(n) or \Sp(2n, \mathbb{R}), where the action preserves the decomposition: the representation space \Lambda^k \mathbb{R}^n (space of k-forms, or totally antisymmetric tensors) decomposes into k-invariant and p-invariant components under the induced involution, facilitating the study of invariant bilinear forms and root systems in the representation theory. For instance, contractions of higher-rank tensors (referencing algebraic properties briefly) yield invariant subspaces aligned with k and p, aiding classifications in symmetric spaces.^[24]^[25]

Examples

Rank-Two Case

A rank-two antisymmetric tensor A_{ij} satisfies A_{ij} = -A_{ji} for all indices i, j, and thus defines a skew-symmetric bilinear form on a vector space V by B(\mathbf{u}, \mathbf{v}) = A_{ij} u^i v^j, where the summation convention is used over repeated indices.^[26] This form is representable by a skew-symmetric matrix with zero diagonal entries, as the antisymmetry implies A_{ii} = 0./07:_Spectral_Theory/7.04:_Orthogonality) In an n-dimensional space, the vector space of such rank-two antisymmetric tensors has dimension n(n-1)/2, spanned by the basis elements e_i \wedge e_j for i < j, where \{e_k\} is the standard basis and \wedge denotes the wedge product in the exterior algebra.^[27] Such a tensor corresponds to a 2-form \omega on a manifold, defined pointwise by \omega(X, Y) = A_{ij} X^i Y^j for vector fields X, Y, capturing the antisymmetric multilinear action.^[28] In the context of differential forms, if the exterior derivative satisfies d\omega = 0, then \omega is closed, a property relevant in symplectic geometry and cohomology. A concrete example arises in three dimensions, where the cross product of vectors \mathbf{v} and \mathbf{w} has components (\mathbf{v} \times \mathbf{w})_i = \varepsilon_{ijk} v^j w^k, with \varepsilon_{ijk} the Levi-Civita symbol representing the oriented volume form; this encodes the antisymmetric tensor structure underlying the vector product.^[10]

Higher-Rank Cases

Antisymmetric tensors of rank greater than two, often referred to as fully or totally antisymmetric tensors, form the space of k-linear alternating multilinear maps on a vector space V, which is isomorphic to the k-th exterior power ∧^k V of V.^[29] In this space, a general element can be expressed as a linear combination of basis elements e_{i_1} ∧ ⋯ ∧ e_{i_k}, where {e_i} is a basis for V and the indices satisfy 1 ≤ i_1 < ⋯ < i_k ≤ n for dim(V) = n, ensuring the antisymmetry under index permutations via the relation v_{\sigma(1)} ∧ ⋯ ∧ v_{\sigma(k)} = \operatorname{sgn}(\sigma) v_1 ∧ ⋯ ∧ v_k for any permutation σ.^[29] The dimension of ∧^k V is the binomial coefficient \binom{n}{k}, reflecting the number of independent components needed to specify such a tensor in n dimensions.^[29] These tensors can be contracted with vectors to produce lower-rank antisymmetric tensors through the interior product operation, which acts as a derivation and reduces the rank by one while preserving antisymmetry; for a vector X and k-form ω, the interior product ι_X ω is a (k-1)-form satisfying ι_X (α ∧ β) = (ι_X α) ∧ β + (-1)^{\deg α} α ∧ (ι_X β).^[30] In an oriented vector space equipped with a metric and volume form, the Hodge dual provides a duality between k-forms and (n-k)-forms. For a basis element e_{i_1} ∧ ⋯ ∧ e_{i_k}, the Hodge dual is given by

*(e_{i_1} ∧ ⋯ ∧ e_{i_k}) = \operatorname{sgn}(\sigma) \, e_{j_1} ∧ ⋯ ∧ e_{j_{n-k}},

where {j_1, ..., j_{n-k}} are the indices complementary to {i_1, ..., i_k} (i.e., the remaining basis indices ordered increasingly), and σ is the permutation that sorts the combined index set into increasing order, with \operatorname{sgn}(\sigma) ensuring orientation consistency.^[30] While fully antisymmetric tensors vanish upon any index repetition, partially antisymmetric tensors exhibit antisymmetry only in specific subsets of indices. For instance, a rank-4 tensor T_{ijkl} might be antisymmetric solely in the first two indices (T_{ij kl} = -T_{ji kl}) but symmetric in the last two (T_{ij kl} = T_{ij lk}), as in partial antisymmetrization denoted by brackets over selected indices, such as T_{[ij]kl}.^[31] Another example is the Riemann curvature tensor in differential geometry, which is antisymmetric in the first two indices but has additional symmetries in others.^[31]

Applications

In Physics

In physics, antisymmetric tensors play a central role in describing fundamental fields and particle statistics. A prominent example is the electromagnetic field tensor F_{\mu\nu} in special relativity, which is a rank-two antisymmetric tensor encoding both the electric and magnetic fields. Its components relate to the three-vectors \mathbf{E} and \mathbf{B} via E_i = F_{0i} and B_i = \frac{1}{2} \epsilon_{ijk} F^{jk} (in the mostly minus metric signature with c=1), where \epsilon_{ijk} is the Levi-Civita symbol. This formulation unifies the electric and magnetic fields into a single Lorentz-covariant object, facilitating the relativistic description of electromagnetism.^[32] Maxwell's equations take a compact tensor form using F_{\mu\nu}: the inhomogeneous equations are \partial_\mu F^{\mu\nu} = J^\nu, expressing the divergence of the field due to the four-current J^\nu, while the homogeneous equations are \partial_\mu {}^*F^{\mu\nu} = 0, where {}^*F^{\mu\nu} = \frac{1}{2} \epsilon^{\mu\nu\rho\sigma} F_{\rho\sigma} is the Hodge dual of F_{\mu\nu}. These equations capture the dynamics of electromagnetic waves and forces on charges in a covariant manner, invariant under Lorentz transformations. In general relativity extended with torsion, such as Einstein-Cartan theory, antisymmetric tensors appear in the torsion tensor T^\lambda_{\mu\nu}, which is antisymmetric in the lower indices \mu\nu and represents the antisymmetric part of the affine connection. The torsion contributes to the stress-energy tensor through coupling to the intrinsic spin of matter, introducing an antisymmetric component that modifies the symmetric energy-momentum tensor of standard general relativity; this allows for a geometric interpretation of spin-gravity interactions without altering the metric compatibility.^[33] In quantum mechanics, antisymmetric tensors underpin the description of identical fermions, where the multi-particle wavefunction must be totally antisymmetric under particle exchange to satisfy the Pauli exclusion principle. For non-interacting fermions, the ground-state wavefunction is constructed as a Slater determinant, a rank-N antisymmetric tensor built from single-particle orbitals, ensuring antisymmetry and providing a basis for Hartree-Fock approximations in many-body theory.^[34]

In Geometry and Topology

In geometry and topology, antisymmetric tensors are fundamental as differential forms on smooth manifolds. A differential k-form \omega on a manifold M is a smooth section of the k-th exterior power of the cotangent bundle, \wedge^k T^*M, representing a totally antisymmetric covariant tensor field of contravariant rank k.^[35] This structure ensures that \omega(X_1, \dots, X_k) = \operatorname{sgn}(\sigma) \omega(X_{\sigma(1)}, \dots, X_{\sigma(k)}) for any permutation \sigma, where X_i are vector fields on M.^[35] Differential forms provide a coordinate-independent framework for integration and differentiation on manifolds, capturing geometric and topological features without reliance on a specific basis.^[30] The exterior derivative d: \Omega^k(M) \to \Omega^{k+1}(M) extends the notion of differentiation to antisymmetric tensors while preserving their alternating property. For a k-form \omega and vector fields X_0, \dots, X_k on M, it is defined by

d\omega(X_0, \dots, X_k) = \sum_{i=0}^k (-1)^i X_i \bigl( \omega(X_0, \dots, \hat{X}_i, \dots, X_k) \bigr) + \sum_{0 \leq i < j \leq k} (-1)^{i+j} \omega\bigl( [X_i, X_j], X_0, \dots, \hat{X}_i, \dots, \hat{X}_j, \dots, X_k \bigr),

where [\cdot, \cdot] denotes the Lie bracket and \hat{\cdot} indicates omission.^[35] This operator satisfies d^2 = 0, enabling the construction of de Rham cohomology groups H^k_{dR}(M) = \ker d / \operatorname{im} d, which are isomorphic to the singular cohomology of M and classify topological invariants.^[35] The antisymmetry of forms ensures that d\omega is well-defined and alternating, as the Lie bracket terms account for non-commutativity of vector fields.^[36] A key application is Stokes' theorem, which relates integration of forms to their boundaries and underpins many topological computations. For a compact oriented (k+1)-dimensional manifold M with boundary \partial M and a k-form \omega with compact support,

\int_M d\omega = \int_{\partial M} \omega.

^[30] This generalizes classical integral theorems (such as the fundamental theorem of calculus) to higher dimensions and holds for manifolds without boundary by taking M as a cycle in a larger space.^[30] The theorem relies on the antisymmetric tensor structure to define oriented integration consistently across charts.^[30] Antisymmetric tensors also feature prominently in characteristic classes, which obstruct the existence of flat connections and encode bundle topology. For a complex vector bundle E \to M with connection whose curvature is the \mathfrak{gl}(n,\mathbb{C})-valued 2-form \Omega, the Chern classes c_i(E) \in H^{2i}(M) are represented by the closed (2i)-forms from the Chern-Weil homomorphism, such as the total Chern form c(E) = \det\left(I + \frac{i}{2\pi} \Omega\right) = 1 + c_1(E) + \cdots + c_n(E).^[37] The antisymmetric nature of \Omega as a Lie algebra-valued 2-form ensures these representatives are closed (dc(E) = 0) via the Bianchi identity D\Omega = 0, where D is the covariant exterior derivative.^[37] Similarly, Pontryagin classes p_i(E) \in H^{4i}(M) for a real vector bundle arise from 4-forms like p_1(E) = -\frac{1}{8\pi^2} \operatorname{Tr}(\Omega \wedge \Omega), leveraging the wedge product of antisymmetric forms to produce even-degree closed invariants.^[37] These classes, independent of the choice of connection, use the alternating tensor algebra to define topological obstructions in bundles over manifolds.^[37]

References

[1]
[PDF] 6 Tensors
A tensor is an object with any number of indices, generalizing vectors and matrices. Its components transform in a specific way, like a vector (rank 1) or ...
[2]
https://www.sciencedirect.com/science/article/pii/B9780128010006000080
[3]
Tensors and pseudo-tensors - Richard Fitzpatrick
The totally antisymmetric tensor is the prototype pseudo-tensor, and is, of course, conventionally defined with respect to a right-handed spatial coordinate ...
[4]
The Feynman Lectures on Physics Vol. II Ch. 31: Tensors
We say “by accident,” because it happens only in three dimensions. In four dimensions, for instance, an antisymmetric tensor of the second rank has up to six ...
[5]
https://www.sciencedirect.com/science/article/pii/B9780124077003000016
[6]
https://link.springer.com/content/pdf/10.1007/978-3-662-69412-1_3.pdf
[7]
[PDF] TENSOR ANALYSIS - II
May 17, 2021 · (b) Antisymmetric tensors or skew symmetric tensors. A tensor, whose each component alters in sign but not in magnitude when two contravariant ...
[8]
[PDF] Multilinear Mappings and Tensors - UCSD CSE
an antisymmetric tensor obeys Aбй = -Aйб. These definitions can also be extended to include contravariant tensors, although we shall have little need to do so.
[9]
[PDF] introduction to the arithmetic theory of quadratic forms - Yale Math
pV qq is the binomial coefficient. ˆm. 𝑛. ˙ for 0 ď 𝑛 ď m, and dimension. 0 for 𝑛 ą m. We define the symmetric algebra SympV q as 'пě0 Symп. pV q. Note that ...
[10]
[PDF] 1 Inner products and norms - Princeton University
Feb 9, 2016 · is the symmetric part of A and C := A−AT. 2 is the anti-symmetric part of A. Notice that xT Cx = 0 for any x ∈ Rn. Example: The matrix. M ...Missing: tensor | Show results with:tensor
[11]
[PDF] c FW Math 321, Mar 5, 2004 Tensor Product and Tensors The tensor ...
It is antisymmetric if it is equal to minus its transpose, i.e. if T = −TT . Any tensor can be decomposed into a symmetric part and an antisymmetric part. T ...
[12]
[PDF] Lecture V: Tensor algebra in flat spacetime
You know in linear algebra that some matrices M can be symmetric (Mαβ = Mβα) or antisymmetric (Mαβ = −Mβα). The same concept applies to tensors of rank 2 and ...
[13]
[PDF] A Primer on Index Notation
Aug 28, 2006 · (c) Any arbitrary tensor T may be decomposed into the sum of a sym- metric tensor (denoted T(ij)) and an antisymmetric tensor (denoted. T[ij]).
[14]
[PDF] 7 Tensors
A tensor is symmetric if Tµν = Tνµ , or is antisymmetric if Tµν = −Tνµ . Obviously, the diagonal elements of an antisymmetric tensor are all zero. ∗ For a ...
[15]
[PDF] Introduction to Tensor Calculus
• anti-symmetric contravariant second-order tensor tµν. An anti-symmetric contravariant second order tensor can be made from two vectors according to. tµν ...
[16]
[PDF] 3. Introducing Riemannian Geometry - DAMTP
Clearly the torsion tensor is anti-symmetric in the lower two indices. Tρ. µν = -Tρ. νµ. Connections which are symmetric in the lower indices, so Γρ. µν = Γρ.<|control11|><|separator|>
[17]
Lecture Notes on General Relativity - S. Carroll
The Riemann tensor, with four indices, naively has n4 independent components in an n-dimensional space. In fact the antisymmetry property (3.64) means that ...
[18]
[PDF] Riemannian Manifolds: An Introduction to Curvature
This book is a textbook for a graduate course on Riemannian geometry, focusing on the geometric meaning of curvature and the main technical tools.
[19]
[PDF] Multilinear Algebra - Alexander Rhys Duncan
Jan 23, 2023 · Since the deter- minant is an alternating multilinear map on the columns of a square matrix, we see that f is an alternating multilinear map ...
[20]
[PDF] Some multilinear algebra
Jan 25, 2020 · We now describe the data determining an alternating multilinear map. Lemma 1.7. If (ei i ∈ I) is a basis of V , then α ∈ Multk(V ;W) is ...
[21]
[PDF] Multilinear Algebra and Tensor Symmetries
Aug 28, 2011 · Alternation operator Alt : V⊗k → V⊗k. Alt(u1 ⊗···⊗ uk) = 1 k! P s∈Sk sgn(s)σ(s)(u1 ⊗···⊗ uk). Properties: •. Alt2 = Alt ( projection ...
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[PDF] MULTILINEAR ALGEBRA 1. Tensor and Symmetric Algebra Let k be ...
A multilinear map f : Mn → N is said to be alternating if it vanishes whenever two arguments are equal. It is not hard to see that it suffices to consider ...
[23]
[PDF] Introduction to Representations of GL(n) - Theorem of the Day
This representation of GL(n) is called the antisymmetric, or exterior, square of the standard representation, written /2 GL(n).
[24]
Determinants and Volumes
In this section we give a geometric interpretation of determinants, in terms of volumes. This will shed light on the reason behind three of the four defining ...
[25]
https://dr.lib.iastate.edu/bitstreams/0db4006b-a6b9-4208-91b7-6d8d17ba31b0/download
[26]
[PDF] Chapter 11 S and Tensor Representations - Rutgers Physics
These are the totally symmetric and totally antisymmetric parts of w, but it is not all of w. For example, suppose w112 = w121 = 1, w211 = −2, all other.
[27]
[PDF] CHAPTER VI Structure Theory of Semisimple Groups
Any two Cartan involutions of g0 are conjugate by an inner automorphism. The Cartan decomposition generalizes the decomposition of a classical matrix Lie ...
[28]
[PDF] Classification of Real Forms of Semisimple Lie Algebras
We have an orthogonal decomposition gc = (h ∩ gc) ⊕. 0 α∈R+. (gα ⊕ g−α) ∩ gc. Moreover, the Killing form is clearly negative definite on h ∩ gc, since.
[29]
[PDF] Lie algebra decompositions with applications to quantum dynamics
The special orthogonal Lie algebra so(n) has two types of Cartan decompositions labeled by BI and BDI, and the symplectic Lie algebra sp(n) has two types of ...
[30]
Bilinear, Quadratic, and Multilinear Forms
Lemma 3.4. Every bilinear form 𝐵 is uniquely the sum 𝐵 = 𝐵+ + 𝐵− of a symmetric and an antisymmetric form. Proof. 𝐵± are given by 𝐵+(𝑣1,𝑣2) = 𝐵(𝑣1,𝑣2) + 𝐵(𝑣2,𝑣 ...
[31]
[PDF] Lecture III: Tensor calculus and electrodynamics in flat spacetime
Oct 8, 2012 · In general, in n-dimensional spacetime, you should be able to show that a bivector has n(n − 1)/2 independent components, and a trivector has n( ...
[32]
[PDF] the symplectic integrability condition - Cornell Mathematics
Sep 3, 2008 · A 2-form on M is a collection ω = {ωp | p ∈ M} of skew-symmetric bilinear forms, one for each tangent space of M. Saying that ω is non ...
[33]
[PDF] EXTERIOR POWERS 1. Introduction Let R be a commutative ring ...
Exterior powers (Λk(M)) are a quotient module of M⊗k, related to alternating multilinear functions, and are important in geometry and linear algebra.
[34]
[PDF] Differential Forms - MIT Mathematics
Feb 1, 2019 · the kth order exterior power of T⋆. p U. (It turns out ... Suppose that we are given a k1-tensor T1, a k2-tensor T2, and a k3-tensor.
[35]
[PDF] A Gentle Introduction to Tensors - Electrical & Systems Engineering
May 27, 2014 · Every inner product space possesses an orthogonal basis and every Euclidean space possesses an orthonormal basis. These are well known results ...
[36]
[PDF] arXiv:0810.4246v1 [physics.class-ph] 23 Oct 2008
Oct 23, 2008 · The F0α terms give the components of the electric field, and the Fαβ terms lead to the components of the magnetic field. This procedure of ...
[37]
On a Completely Antisymmetric Cartan Torsion Tensor - gr-qc - arXiv
Jun 25, 2012 · Title:On a Completely Antisymmetric Cartan Torsion Tensor ; Comments: 11 pages ; Subjects: General Relativity and Quantum Cosmology (gr-qc); High ...
[38]
[PDF] Machine Learning Wavefunction - arXiv
Feb 28, 2022 · Figure 1: Graphical representation of a Slater determinant (left) and of a 1D lattice (right). ... antisymmetric wavefunction, and an easy ...
[39]
[PDF] arXiv:1206.3323v1 [math-ph] 20 May 2012
May 20, 2012 · The first requirement implies that differential forms have to be tensors, objects whose physical manifestation does not change under coordinate ...
[40]
[PDF] Multilinear algebra, differential forms and Stokes' theorem
that dω = 0, i.e ω is closed. On the other hand it is not exact. Indeed, let ... Work out an explicit expression for a primitive of a closed 2-form α = P dy∧dz+.
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[PDF] A Note on Characteristic Classes - arXiv
This paper studies the relationship between the sections and the Chern or. Pontrjagin classes of a vector bundle by the theory of connection. Our results are ...