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Antiunitary operator
Antiunitary operator
Antiunitary operator
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In mathematics, an antiunitary transformation is a bijective antilinear map

between two complex Hilbert spaces such that

for all and in , where the horizontal bar represents the complex conjugate. If additionally one has then is called an antiunitary operator.

Antiunitary operators are important in quantum mechanics because they are used to represent certain symmetries, such as time reversal.[1] Their fundamental importance in quantum physics is further demonstrated by Wigner's theorem.

Invariance transformations

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In quantum mechanics, the invariance transformations of complex Hilbert space leave the absolute value of scalar product invariant:

for all and in .

Due to Wigner's theorem these transformations can either be unitary or antiunitary.

Geometric Interpretation

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Congruences of the plane form two distinct classes. The first conserves the orientation and is generated by translations and rotations. The second does not conserve the orientation and is obtained from the first class by applying a reflection. On the complex plane these two classes correspond (up to translation) to unitaries and antiunitaries, respectively.

Properties

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  • holds for all elements of the Hilbert space and an antiunitary .
  • When is antiunitary then is unitary. This follows from
  • For unitary operator the operator , where is complex conjugation (with respect to some orthogonal basis), is antiunitary. The reverse is also true, for antiunitary the operator is unitary.
  • For antiunitary the definition of the adjoint operator is changed to compensate the complex conjugation, becoming
  • The adjoint of an antiunitary is also antiunitary and (This is not to be confused with the definition of unitary operators, as the antiunitary operator is not complex linear.)

Examples

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  • The complex conjugation operator is an antiunitary operator on the complex plane.
  • The operator where is the second Pauli matrix and is the complex conjugation operator, is antiunitary. It satisfies .

Decomposition of an antiunitary operator into a direct sum of elementary Wigner antiunitaries

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An antiunitary operator on a finite-dimensional space may be decomposed as a direct sum of elementary Wigner antiunitaries , . The operator is just simple complex conjugation on

For , the operator acts on two-dimensional complex Hilbert space. It is defined by

Note that for

so such may not be further decomposed into 's, which square to the identity map.

Note that the above decomposition of antiunitary operators contrasts with the spectral decomposition of unitary operators. In particular, a unitary operator on a complex Hilbert space may be decomposed into a direct sum of unitaries acting on 1-dimensional complex spaces (eigenspaces), but an antiunitary operator may only be decomposed into a direct sum of elementary operators on 1- and 2-dimensional complex spaces.

References

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See also

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Revisions and contributorsEdit on WikipediaRead on Wikipedia

Antiunitary operator

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An antiunitary operator on a complex H\mathcal{H} is an antilinear A:H→HA: \mathcal{H} \to \mathcal{H} that preserves the inner product up to complex conjugation, satisfying ⟨Aψ∣Aϕ⟩=⟨ψ∣ϕ⟩∗\langle A\psi | A\phi \rangle = \langle \psi | \phi \rangle^* for all ψ,ϕ∈H\psi, \phi \in \mathcal{H}. Antilinearity means A(cψ+ϕ)=c∗Aψ+AϕA(c\psi + \phi) = c^* A\psi + A\phi for any complex scalar cc and vectors ψ,ϕ\psi, \phi. Every such operator can be expressed as A=UKA = UK, where UU is a on H\mathcal{H} and KK denotes complex conjugation with respect to some fixed of H\mathcal{H}. Antiunitary operators arise prominently in the representation of symmetries in , as established by , which states that any bijective map on the CP(H)\mathbb{CP}(\mathcal{H}) preserving transition probabilities ∣⟨ψ∣ϕ⟩∣2|\langle \psi | \phi \rangle|^2 lifts to either a unitary or an antiunitary operator on H\mathcal{H}. This theorem, originally formulated by , underscores that physical symmetries preserving the absolute value of inner products are implemented by these two classes of operators, with antiunitary ones corresponding to symmetries involving an "orientation reversal" in the complex structure of the space. A canonical example is the time-reversal operator , which is antiunitary and implements time-reversal invariance in by mapping states forward in time to their counterparts backward in time, satisfying ΘHΘ−1=H\Theta H \Theta^{-1} = H for time-independent Hamiltonians HH. The antilinearity of Θ\Theta arises from the need to complex-conjugate wave functions to reverse momenta and preserve probabilities under time reversal, as ΘpΘ−1=−p\Theta p \Theta^{-1} = -p for the pp. Antiunitary operators also feature in other , such as charge conjugation combined with parity in fermionic systems or particle-hole symmetry in superconductors, influencing properties, Kramers degeneracy, and topological phases.

Definition and Mathematical Foundations

Formal Definition

An antiunitary operator UU on a complex Hilbert space H\mathcal{H} is a bijective U:H→HU: \mathcal{H} \to \mathcal{H} that preserves the inner product up to complex conjugation, satisfying ⟨Uψ,Uϕ⟩=⟨ψ,ϕ⟩∗\langle U\psi, U\phi \rangle = \langle \psi, \phi \rangle^{*} for all ψ,ϕ∈H\psi, \phi \in \mathcal{H}, where ∗^{*} denotes complex conjugation. This condition ensures that antiunitary operators maintain the geometric structure of the space, including the norm, since ∥Uψ∥2=∥ψ∥2\|U\psi\|^2 = \|\psi\|^2. Antilinearity distinguishes these operators from linear ones: for scalars α,β∈C\alpha, \beta \in \mathbb{C}, U(αψ+βϕ)=α∗Uψ+β∗Uϕ.U(\alpha \psi + \beta \phi) = \alpha^{*} U\psi + \beta^{*} U\phi. Unlike linearity, which applies scalars directly without conjugation, antilinearity conjugates the coefficients, reflecting the involvement of complex conjugation in the inner product preservation. General antilinear operators, by contrast, do not necessarily satisfy the inner product condition and thus fail to preserve norms or the up to conjugation. Antiunitary operators exist and can be constructed as the composition of a and a complex conjugation operator KK, where KK is defined with respect to an and extended antilinearly.

Relation to Unitary Operators

Antiunitary operators bear a close structural relationship to , differing primarily in their linearity properties. Every antiunitary operator UU on a complex can be decomposed as U=VJU = V J, where VV is a satisfying V†V=IV^\dagger V = I and thus preserving the inner product ⟨Vψ∣Vϕ⟩=⟨ψ∣ϕ⟩\langle V\psi | V\phi \rangle = \langle \psi | \phi \rangle, while JJ denotes the complex conjugation operator with respect to a fixed {en}\{e_n\}, defined by Jψ=∑n⟨en∣ψ⟩∗enJ\psi = \sum_n \langle e_n | \psi \rangle^* e_n. This JJ is antilinear, satisfies J2=IJ^2 = I, and in the position representation acts as Jψ(x)=ψ∗(x)J \psi(x) = \psi^*(x). The decomposition highlights that antiunitaries extend unitaries by incorporating an antilinear conjugation element, enabling the representation of symmetries like time reversal that involve complex conjugation. To verify that U=VJU = V J satisfies the antiunitary condition, note first that UU is antilinear as the composition of the linear VV and antilinear JJ: for scalars a,b∈Ca, b \in \mathbb{C}, U(aψ+bϕ)=VJ(aψ+bϕ)=V(a∗Jψ+b∗Jϕ)=a∗VJψ+b∗VJϕ=a∗Uψ+b∗UϕU(a\psi + b\phi) = V J(a\psi + b\phi) = V(a^* J\psi + b^* J\phi) = a^* V J\psi + b^* V J\phi = a^* U\psi + b^* U\phi. For the inner product preservation, antiunitaries satisfy ⟨Uψ∣Uϕ⟩=⟨ψ∣ϕ⟩∗\langle U\psi | U\phi \rangle = \langle \psi | \phi \rangle^*. Substituting the decomposition yields ⟨Uψ∣Uϕ⟩=⟨VJψ∣VJϕ⟩=⟨Jψ∣Jϕ⟩\langle U\psi | U\phi \rangle = \langle V J\psi | V J\phi \rangle = \langle J\psi | J\phi \rangle since VV is unitary. Now, ⟨Jψ∣Jϕ⟩=∑n⟨Jψ∣en⟩⟨en∣Jϕ⟩=∑n⟨ψ∣en⟩∗⟨en∣ϕ⟩∗=(∑n⟨en∣ψ⟩⟨ψ∣en⟩∗)∗=⟨ψ∣ϕ⟩∗\langle J\psi | J\phi \rangle = \sum_n \langle J\psi | e_n \rangle \langle e_n | J\phi \rangle = \sum_n \langle \psi | e_n \rangle^* \langle e_n | \phi \rangle^* = \left( \sum_n \langle e_n | \psi \rangle \langle \psi | e_n \rangle^* \right)^* = \langle \psi | \phi \rangle^*, confirming the condition. This relation underscores the conjugate-preserving nature of antiunitaries compared to the direct preservation by unitaries. The proof of the decomposition's existence proceeds by fixing an orthonormal basis and defining JJ as above, then setting V=UJV = U J (noting J−1=JJ^{-1} = J). Linearity of VV follows from the antilinearity of both UU and JJ, yielding a linear operator. Unitarity is established via ⟨Vψ∣Vϕ⟩=⟨UJψ∣UJϕ⟩=⟨Jψ∣Jϕ⟩∗=⟨ψ∣ϕ⟩∗∗=⟨ψ∣ϕ⟩\langle V\psi | V\phi \rangle = \langle U J\psi | U J\phi \rangle = \langle J\psi | J\phi \rangle^* = \langle \psi | \phi \rangle^{**} = \langle \psi | \phi \rangle, as derived earlier. Alternatively, a polar decomposition for antilinear operators K=ACK = A C, where CC is conjugation and AA linear, can be applied; for antiunitary UU, AA is unitary, yielding the form directly. The decomposition is unique up to phase factors: if U=VJ=V′J′U = V J = V' J', then V′=VeiθV' = V e^{i\theta} and J′=e−iθJJ' = e^{-i\theta} J for some real θ\theta, though the latter adjustment preserves the conjugation property only in adapted bases. The operator JJ is not unique across all bases but is fixed once a standard is chosen, ensuring the representation is well-defined in that frame. This basis dependence reflects the interplay between the complex structure of the and the choice of representation for conjugation.

Core Properties

Algebraic Properties

Antiunitary operators on a complex Hilbert space H\mathcal{H} satisfy the condition U†U=IU^\dagger U = I and UU†=IU U^\dagger = I, where the U†U^\dagger is defined such that ⟨Uψ,ϕ⟩=⟨ψ,U†ϕ⟩∗\langle U \psi, \phi \rangle = \langle \psi, U^\dagger \phi \rangle^* for all ψ,ϕ∈H\psi, \phi \in \mathcal{H}. This condition ensures that antiunitary operators preserve the norm of vectors, i.e., ∥Uψ∥=∥ψ∥\|U \psi\| = \|\psi\| for all ψ∈H\psi \in \mathcal{H}, and more generally, they preserve inner products up to complex conjugation: ⟨Uψ,Uϕ⟩=⟨ψ,ϕ⟩∗\langle U \psi, U \phi \rangle = \langle \psi, \phi \rangle^*. As a consequence, the inverse of an antiunitary operator coincides with its , U−1=U†U^{-1} = U^\dagger. The defining feature of an antiunitary operator is its antilinearity: for scalars α,β∈C\alpha, \beta \in \mathbb{C} and vectors ψ,ϕ∈H\psi, \phi \in \mathcal{H}, U(αψ+βϕ)=α∗Uψ+β∗Uϕ,U(\alpha \psi + \beta \phi) = \alpha^* U \psi + \beta^* U \phi, where ∗^* denotes complex conjugation. This antilinearity has direct implications for compositions. The composition of two antiunitary operators UU and VV is a , as the antilinear maps compose to yield a that preserves inner products without conjugation: (UV)†(UV)=I(UV)^\dagger (UV) = I. Conversely, the composition of an antiunitary operator with a remains antiunitary. A fundamental identity arising from antilinearity is that the square of an antiunitary operator U2U^2 is unitary. In specific representations, such as the time-reversal operator for systems with , U2=(−1)2jIU^2 = (-1)^{2j} I, where jj is the ; this yields U2=IU^2 = I for spin and U2=−IU^2 = -I for spin. The collection of all unitary and antiunitary operators on H\mathcal{H} forms a group under composition, known as the unitary-antiunitary group UA(H)UA(\mathcal{H}), with the unitary operators comprising the connected component containing the identity. Within this group, the antiunitary operators form a of the unitary subgroup, reflecting their role in extending linear symmetries to include conjugation-like operations. Due to antilinearity, the set of antiunitary operators does not form a over C\mathbb{C}, as would violate the preservation of inner products up to conjugation. All antiunitary operators on a are bounded linear (or antilinear) operators with ∥U∥=1\|U\| = 1, following directly from the condition and the completeness of H\mathcal{H}.

Spectral Properties

Antiunitary operators, due to their antilinear nature, do not possess eigenvalues in the conventional linear sense, as the set of solutions to the eigenvector equation Uψ=λψU \psi = \lambda \psi does not form a vector subspace closed under complex . Applying UU to both sides of the equation yields U2ψ=λ∗Uψ=λ∗λψU^2 \psi = \lambda^* U \psi = \lambda^* \lambda \psi, implying ∣λ∣2=1|\lambda|^2 = 1 if U2=IU^2 = I, which is typical for many representations; more generally, U2U^2 is unitary, constraining the possible λ\lambda further. For real eigenvalues, only λ=±1\lambda = \pm 1 are possible, corresponding to eigenspaces where Uψ=ψU \psi = \psi (fixed points) or Uψ=−ψU \psi = -\psi (anti-fixed points). These real eigenvalues arise in one-dimensional invariant subspaces under Wigner's normal form, where the operator leaves basis vectors invariant up to a phase that can be chosen to yield ±1\pm 1. If U2=−IU^2 = -I, as in time-reversal for half-integer spin systems, no such real eigenvalues exist, leading to at least twofold degeneracy without eigenvectors. For non-real λ\lambda with ∣λ∣=1|\lambda| = 1, eigenvectors occur in conjugate pairs: if Uψ=λψU \psi = \lambda \psi, then there exists ϕ\phi such that Uϕ=λ∗ϕU \phi = \lambda^* \phi, typically with ϕ\phi related to ψ\psi via the action of UU (e.g., ϕ∝Uψ/λ∗\phi \propto U \psi / \lambda^*) and satisfying ⟨ψ,ϕ⟩=0\langle \psi, \phi \rangle = 0. These pairs span two-dimensional invariant subspaces in Wigner's , where the antiunitary acts by swapping basis vectors with phase factors e±iθe^{\pm i \theta}, ensuring the spectrum lies on the unit circle with conjugation . Unlike unitary operators, antiunitary operators lack a full providing a diagonalization over the entire , as antilinearity prevents a complete set of orthogonal eigenvectors. Instead, the space decomposes into orthogonal direct sums of one- and two-dimensional invariant subspaces, restricting spectral analysis to these finite-dimensional blocks. This norm-preserving property ∥Uψ∥=∥ψ∥\| U \psi \| = \| \psi \| for all ψ\psi confirms the spectrum's location on the unit circle, with the pairing mechanism enforcing conjugate symmetry even in the absence of traditional eigenvalues.

Physical Interpretations and Applications

Time-Reversal Invariance

In , time-reversal invariance is implemented by an antiunitary operator TT, which ensures that the laws of physics remain unchanged under the reversal of time. For a time-independent Hamiltonian HH, the condition for time-reversal invariance is THT−1=HT H T^{-1} = H, meaning the operator conjugates the Hamiltonian to itself while preserving the dynamics in the reversed time direction. This antiunitarity arises because time reversal must reverse the direction of time derivatives in the Schrödinger equation, leading to the property TiT−1=−iT i T^{-1} = -i, where ii is the ; this follows directly from the antilinear nature of TT, as T(cψ)=c∗TψT (c \psi) = c^* T \psi for complex cc, and thus conjugates scalars like ii to −i-i. The action of TT on quantum states in the Schrödinger picture maps a state evolving forward in time to its time-reversed counterpart, typically expressed as T∣ψ(t)⟩=T∣ψ(−t)⟩∗T |\psi(t)\rangle = T |\psi(-t)\rangle^*, where the asterisk denotes complex conjugation in a basis where TT acts as conjugation. This transformation preserves transition probabilities, ∣⟨ϕ∣ψ⟩∣2=∣⟨Tϕ∣Tψ⟩∣2|\langle \phi | \psi \rangle|^2 = |\langle T\phi | T\psi \rangle|^2, but conjugates the phases of the states, reflecting the antiunitary structure that inverts the sign of phase angles under time reversal. According to an extension of Wigner's theorem, symmetries of the projective Hilbert space (ray space) that preserve inner product magnitudes can be represented by either unitary or antiunitary operators on the full Hilbert space; time reversal falls into the antiunitary category because a unitary operator would fail to reverse momenta and phases appropriately. The consequences for observables depend on their time parity: time-even operators, such as position r\mathbf{r}, satisfy TrT−1=rT \mathbf{r} T^{-1} = \mathbf{r} and thus commute with TT, while time-odd operators, such as momentum p\mathbf{p} and angular momentum L=r×p\mathbf{L} = \mathbf{r} \times \mathbf{p}, satisfy TOT−1=−OT \mathbf{O} T^{-1} = -\mathbf{O} and anticommute with TT. This distinction ensures that velocities and rotations reverse under time reversal, consistent with classical intuitions extended to quantum systems. Eugene Wigner introduced the antiunitary time-reversal operator in 1931, particularly in the context of spin-1/2 systems, where he demonstrated its necessity for maintaining symmetry in atomic spectra. A key property is that T2=(−1)2sT^2 = (-1)^{2s}, where ss is the spin quantum number; this yields T2=+1T^2 = +1 for integer spin (bosons) and T2=−1T^2 = -1 for half-integer spin (fermions), leading to Kramers' degeneracy in the latter case under time-reversal invariance.

Geometric and Symmetry Interpretations

Antiunitary operators possess a profound geometric interpretation in the context of Hilbert spaces, where they act as anti-isometries that reverse the orientation of the while preserving the metric structure up to complex conjugation. Unlike unitary operators, which induce orientation-preserving transformations akin to rotations, antiunitaries implement orientation-reversing isometries, such as reflections, in the geometry of the state space. This reflective property stems from their anti-linear nature, which effectively conjugates the complex phases, mirroring the action of complex conjugation in the that flips the imaginary axis across the real line. In the framework of symmetry groups GG acting on quantum systems, antiunitary elements correspond to orientation-reversing transformations in the projective Hilbert space, extending the unitary representations to include improper isometries that are crucial for capturing full symmetry structures, such as those involving spatial reflections or inversions. These operators ensure invariance of physical observables under such transformations by mapping states in a way that conjugates inner products, thereby maintaining the probabilistic interpretation while inverting the geometric orientation. For instance, in systems with discrete symmetries, antiunitaries facilitate the implementation of group elements that would otherwise be inaccessible through unitary means alone. Charge conjugation, which interchanges particles and antiparticles, is formulated as an antiunitary operator in certain quantum mechanical contexts, particularly in approaches or when considering the interplay with other discrete symmetries. This antiunitary character arises because charge conjugation involves complex conjugation of field operators, leading to anti-linear effects on the states. In the context of the TCP , the combined charge conjugation (C), parity (P), and time reversal (T) transformation is antiunitary overall, since C and P are unitary while T is antiunitary, guaranteeing the invariance of relativistic quantum field theories under this full . Antiunitary operators also play a role in extending unitary groups, such as U(n)U(n), to broader structures that incorporate conjugation-like operations, often referred to as pseudo-unitary extensions in the presence of indefinite metrics or enlargements. These extensions generate groups that include both orientation-preserving and reversing elements, allowing for a complete description of operations that mix unitary and antiunitary actions. In finite-dimensional cases, such as systems, the action of antiunitaries on the manifests as reflections through the real axis, effectively flipping the sphere's orientation and providing a visualizable geometric counterpart to the abstract transformations; similar reflections occur in the of coherent states, underscoring the pervasive reflective geometry.

Examples and Constructions

Basic Examples

One of the simplest examples of an antiunitary operator is the complex conjugation operator JJ acting on the L2(R)L^2(\mathbb{R}), defined by (Jf)(x)=f(x)‾(J f)(x) = \overline{f(x)} for a function f∈L2(R)f \in L^2(\mathbb{R}). This operator is antilinear, as J(cf)=c‾(Jf)J(c f) = \overline{c} (J f) for complex cc, and it preserves the inner product up to conjugation: ⟨Jf,Jg⟩=⟨f,g⟩‾\langle J f, J g \rangle = \overline{\langle f, g \rangle}, confirming its antiunitarity. Moreover, J2=IJ^2 = I, the identity operator. In the context of time-reversal symmetry for a spinless particle, the time-reversal operator TT in the position basis simplifies to complex conjugation, so Tψ(x)=ψ(x)‾T \psi(x) = \overline{\psi(x)}. More generally, in the basis, T=JKT = J K, where KK is the momentum reversal operator (implemented via the ), but the position representation reduces to JJ alone since wave functions can be chosen real for spinless systems. This operator satisfies T2=IT^2 = I and reverses velocities while preserving positions. For a particle, the time-reversal operator incorporates the spin degree of freedom and is given by T=iσyKT = i \sigma_y K, where σy\sigma_y is the Pauli y-matrix and KK denotes componentwise complex conjugation in the basis. Acting on a ψ=(αβ)\psi = \begin{pmatrix} \alpha \\ \beta \end{pmatrix}, this yields Tψ=iσy(α‾β‾)=(β‾−α‾)T \psi = i \sigma_y \begin{pmatrix} \overline{\alpha} \\ \overline{\beta} \end{pmatrix} = \begin{pmatrix} \overline{\beta} \\ -\overline{\alpha} \end{pmatrix}, which flips the spin direction while complex conjugating the components. It satisfies T2=−IT^2 = -I, reflecting the fermionic nature of systems, and is antiunitary as it preserves the inner product up to conjugation. A finite-dimensional example occurs in C2\mathbb{C}^2, where the operator U=σxKU = \sigma_x K provides an antiunitary map, with σx=(0110)\sigma_x = \begin{pmatrix} 0 & 1 \\ 1 & 0 \end{pmatrix} and KK the conjugation. This illustrates how antiunitaries in low dimensions can mimic reflections while involving conjugation, and U2=IU^2 = I. In (QED), the charge conjugation operator for Dirac fields acts on a ψ\psi as Cψ=iγ2ψ‾C \psi = i \gamma^2 \overline{\psi}, or equivalently Cψ=iγ2ψ∗C \psi = i \gamma^2 \psi^* in the single-particle description, where γ2\gamma^2 is a Dirac matrix and ∗* denotes complex conjugation. This operator is antiunitary due to the conjugation, interchanging particle and antiparticle states while preserving the Dirac equation's form, and satisfies C2=IC^2 = I in the standard representation.

Decomposition into Elementary Antiunitaries

In , an elementary antiunitary operator, often referred to as a Wigner antiunitary, acts on an space and takes the form U=eiθVKU = e^{i\theta} V K, where VV is a , KK denotes complex conjugation with respect to a chosen , and θ\theta is a real . This normal form arises from the ability to select a basis in which the antiunitary operator simplifies to a phase times a unitary followed by conjugation, capturing the essential structure without loss of generality. A fundamental result in the theory is the decomposition theorem for antiunitary operators on a separable HH: any such operator UU can be expressed as a U=⨁jUjU = \bigoplus_j U_j, where each UjU_j is an elementary Wigner antiunitary acting on a minimal UU- HjH_j. These minimal subspaces are the building blocks, analogous to irreducible subspaces in unitary , and the full space decomposes orthogonally as H=⨁jHjH = \bigoplus_j H_j. The theorem extends to normal antilinear operators more broadly, partitioning HH into Hermitian-diagonalizable parts (with eigenvectors) and even-dimensional parts lacking real eigenvectors but paired via the operator. The proof sketch leverages an extension of to the enlarged algebra generated by both unitary and antiunitary actions. Specifically, for an irreducible corepresentation under this joint action, any intertwining antiunitary operator must be scalar (up to unitary equivalence) or zero, ensuring the minimality and irreducibility of the HjH_j. This irreducibility criterion prevents further nontrivial invariant subspaces, mirroring the unitary case but accounting for the antilinearity through basis-dependent conjugation. The minimal subspaces HjH_j are classified up to equivalence by their and the of Uj2=±IU_j^2 = \pm I. When Uj2=IU_j^2 = I, the representation is of conjugation type, allowing real eigenvalues; in contrast, Uj2=−IU_j^2 = -I implies skew-conjugation type, requiring even ality and leading to Kramers degeneracy, where states cannot be nondegenerate due to under the antiunitary. Multiplicities arise from repeated equivalent blocks in the , determined by the overall of HH. This decomposition framework underpins symmetry classifications in physical systems, particularly for crystal space groups and molecular symmetries incorporating time-reversal, where antiunitaries combine with spatial operations to form magnetic or gray groups. The irreducible types dictate degeneracy patterns and topological invariants in such contexts.

Advanced Extensions

Representations in Hilbert Spaces

In a Hilbert space equipped with an orthonormal basis {en}\{e_n\}, an antiunitary operator UU acts on a vector ψ=∑nψnen\psi = \sum_n \psi_n e_n by first complex-conjugating the coefficients and then applying a unitary transformation represented by the matrix elements Umn=⟨em∣Uen⟩U_{mn} = \langle e_m | U e_n \rangle, yielding Uψ=∑m,nemUmnψn‾U\psi = \sum_{m,n} e_m U_{mn} \overline{\psi_n}. This representation highlights the antilinear nature of UU, as the conjugation step distinguishes it from unitary operators, which act linearly on the coefficients without conjugation. The matrix UmnU_{mn} itself satisfies unitarity conditions adjusted for the conjugation, ensuring ⟨Uψ∣Uϕ⟩=⟨ϕ∣ψ⟩∗\langle U\psi | U\phi \rangle = \langle \phi | \psi \rangle^* for all ψ,ϕ\psi, \phi. In the position basis on L2(Rd)L^2(\mathbb{R}^d), antiunitary operators typically combine complex conjugation KK, defined by (Kf)(x)=f(x)‾(K f)(\mathbf{x}) = \overline{f(\mathbf{x})}, with a . For the time-reversal operator TT acting on a spinless particle, the explicit form is T=KT = K, so (Tf)(x)=f(x)‾(T f)(\mathbf{x}) = \overline{f(\mathbf{x})}; this preserves the position but reverses via TpT−1=−pT \mathbf{p} T^{-1} = -\mathbf{p}, as the conjugation flips the sign of the in the . Unitarity is verified by ∥Tf∥2=∫∣f(x)‾∣2dx=∥f∥2\|T f\|^2 = \int |\overline{f(\mathbf{x})}|^2 d\mathbf{x} = \|f\|^2 and the inner product relation ⟨Tf∣Tg⟩=⟨g∣f⟩∗\langle T f | T g \rangle = \langle g | f \rangle^*. In the basis, the action becomes (Tϕ)(p)=ϕ(−p)‾(T \phi)(\mathbf{p}) = \overline{\phi(-\mathbf{p})}, incorporating a from the that reflects the momentum reversal. For infinite-dimensional Hilbert spaces like L2(Rd)L^2(\mathbb{R}^d), explicit constructions of antiunitaries often factor as U=VKU = V K, where VV is unitary and KK is complex conjugation in the position basis. The time-reversal example T=KT = K satisfies antiunitarity: T(cf)=c‾TfT (c f) = \overline{c} T f for complex cc, and T†T=IT^\dagger T = I with the adjoint defined via the conjugated inner product. Such operators preserve the Hilbert space norm and orthogonality up to conjugation, enabling their role in symmetry transformations without altering the space's structure. In the Bargmann representation, where states of the are holomorphic functions on C\mathbb{C} with the measure e−∣z∣2dμ(z)e^{-|z|^2} d\mu(z), antiunitary operators act as antilinear combinations of differential operators followed by complex conjugation, mapping holomorphic functions to antiholomorphic ones while preserving the inner product up to conjugation. Two antiunitary operators UU and U′U' on a are equivalent if there exists a WW such that U′=WUW−1U' = W U W^{-1}, intertwining their actions and preserving the antiunitary structure through the linearity of WW. This unitary equivalence classifies representations up to basis changes, ensuring that spectral or transformation properties remain invariant.

Antiunitary Groups and Algebras

Antiunitary groups arise in quantum mechanics as extensions of unitary symmetry groups to include orientation-reversing transformations, forming closed subgroups under operator composition where the product of two antiunitary operators is unitary. For a Lie group GG with a normal subgroup G1⊆GG_1 \subseteq G of index 2 (often the identity component), an antiunitary representation maps elements of G∖G1G \setminus G_1 to antiunitary operators on a Hilbert space while preserving the group structure, thus combining unitary and antiunitary operators into a cohesive representation of the full group. Such groups capture full symmetry operations, including those like improper rotations, which reverse orientation and require antiunitarity to preserve probabilities in quantum systems. Incorporating antiunitary operators into Lie algebras presents challenges due to their antilinearity, which disrupts standard linear ; generators may include anti-Hermitian components, but the algebra structure relies on modular operators like the modular conjugation JJ and Hamiltonian Δ\Delta, where one-parameter subgroups generated by antiunitaries relate to boosts or dilations in groups like the affine group Aff(R)\mathrm{Aff}(\mathbb{R}). For instance, in representations of the , antiunitary extensions yield elements tied to time-reversal symmetries, complicating the infinitesimal structure but enabling descriptions of modular theory in von Neumann algebras. Examples of antiunitary groups include the O(n)O(n), which, in the of real representations, decomposes into unitary proper rotations ( +1) and antiunitary improper transformations ( -1), preserving the real structure while accounting for conjugation. More generally, Pin groups in s provide concrete realizations: the Pin(p,q)(p,q) groups are double covers of the orthogonal group O(p,q)O(p,q), generated by unit vectors in the Cl(p,q)\mathrm{Cl}(p,q) with norm ±1\pm 1, where elements act as unitary or antiunitary transformations on spaces, particularly incorporating antiunitarity for time reversal TT via co-representations like T=γ5γ0∗T = \gamma_5 \gamma_0^*. In physics, Pin(1,3)\mathrm{Pin}(1,3) and Pin(3,1)\mathrm{Pin}(3,1) distinguish fermion behaviors under charge conjugation CC, parity PP, and time reversal TT, with antiunitary TT ensuring T2=−1T^2 = -1 for particles. Irreducible representations of antiunitary groups are classified using corepresentations, where the Frobenius-Schur indicator ν(ρ)=∫Gtr⁡(ρ(g2)) dg\nu(\rho) = \int_G \operatorname{tr}(\rho(g^2)) \, dg (over the ) determines the type: ν=1\nu = 1 for real representations (extendable unitarily with J2=1J^2 = 1), ν=−1\nu = -1 for quaternionic (requiring antiunitary JJ with J2=−1J^2 = -1, as in fermionic time reversal), and ν=0\nu = 0 for complex (non-extendable). This indicator, extended to corepresentations, identifies whether an irreducible unitary representation of the unitary subgroup extends irreducibly to the full antiunitary group, crucial for types in . In topological contexts, antiunitary groups connect to K-theory through extensions classifying symmetry-protected phases, such as in topological insulators where time-reversal symmetry (antiunitary with T2=−1T^2 = -1) leads to Z2\mathbb{Z}_2 invariants via quaternionic K-theory (KQ-theory) or real K-theory (KR-theory). The bulk-boundary correspondence maps bulk KQ-classes over momentum space to boundary KO-invariants at time-reversal fixed points, with the Z2\mathbb{Z}_2 index given by the mod 2 parity of Majorana zero modes, distinguishing trivial from nontrivial insulators in class AII.

References

  1. https://arxiv.org/pdf/1301.7639.pdf
  2. https://arxiv.org/pdf/0808.0779.pdf
  3. https://bohr.physics.berkeley.edu/classes/221/1112/notes/timerev.pdf
  4. https://arxiv.org/pdf/0912.2157.pdf
  5. https://arxiv.org/pdf/0810.3536
  6. https://ncatlab.org/nlab/show/antiunitary+operator
  7. https://michaelberryphysics.wordpress.com/wp-content/uploads/2013/07/berry146.pdf
  8. https://scipp.ucsc.edu/~haber/ph215/antilinearops.pdf
  9. https://arxiv.org/pdf/0810.3536.pdf
  10. https://icourse.club/uploads/files/d05a1e3f8ecfe2315de4d8958f9c4a904346d472.pdf
  11. https://www.isibang.ac.in/~gm/gmhomefiles/papers/KRPImprimitivity.pdf
  12. https://arxiv.org/pdf/quant-ph/0209054.pdf
  13. https://pubs.aip.org/aip/jmp/article-pdf/1/5/409/11276242/409_1_online.pdf
  14. https://.org/pdf/quant-ph/0209054.pdf
  15. https://personal.lse.ac.uk/robert49/pdf/Roberts2017a.pdf
  16. https://www.cambridge.org/core/journals/philosophy-of-science/article/three-myths-about-time-reversal-in-quantum-theory/48EFABE33CF7A3C33E7B5D9F172515E5
  17. https://arxiv.org/pdf/1607.07388
  18. https://arxiv.org/abs/2204.14134
  19. http://www.math.caltech.edu/SimonPapers/R12.pdf
  20. https://arxiv.org/abs/1704.01336
  21. https://arxiv.org/abs/1409.3681
  22. https://personal.lse.ac.uk/robert49/teaching/partiii/2020-2021/pdf/cpt.pdf
  23. https://pubs.aip.org/aip/jmp/article-pdf/25/5/1205/19081544/1205_1_online.pdf
  24. https://bingweb.binghamton.edu/~suzuki/QM_Graduate/Time_reversal_I_operator.pdf
  25. https://beckassets.blob.core.windows.net/product/readingsample/448073/9783642054273_excerpt_001.pdf
  26. https://web2.ph.utexas.edu/~vadim/Classes/2022f/CC.pdf
  27. https://projecteuclid.org/journals/communications-in-mathematical-physics/volume-15/issue-4/Projective-unitary-antiunitary-representations-of-locally-compact-groups/cmp/1103841989.pdf
  28. https://link.springer.com/article/10.1007/BF00671657
  29. https://mathoverflow.net/questions/263131/a-question-about-unitary-and-anti-unitary-matrices
  30. http://deferentialgeometry.org/papers/The%20Pin%20Groups%20in%20Physics-%20C%2C%20P%2C%20and%20T.pdf
  31. https://www.math.purdue.edu/~ebkaufma/1510.08001.pdf
  32. https://pubs.aip.org/aip/jmp/article/65/4/043502/3282730/Topological-insulators-and-K-theory
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