Generalized Wigner theorem for non-invertible symmetries

Generalized Wigner theorem for non-invertible symmetries

We establish the conditions under which a conservation law associated with a non-invertible operator may be realized as a symmetry in quantum mechanics. As established by Wigner, all quantum symmetries must be represented by either unitary or antiunitary transformations. Relinquishing an implicit assumption of invertibility, we demonstrate that the fundamental invariance of quantum transition probabilities under the application of symmetries mandates that all non-invertible symmetries may only correspond to projective unitary or antiunitary transformations, i.e., partial isometries. This extends the notion of physical states beyond conventional rays in Hilbert space to equivalence classes in an extended, gauged Hilbert space, thereby broadening the traditional understanding of symmetry transformations in quantum theory. We discuss consequences of this result and explicitly illustrate how, in simple model systems, whether symmetries be invertible or non-invertible may be inextricably related to the particular boundary conditions that are being used.

Introduction. Symmetry is a fundamental organizing principle underlying the laws of nature. At its core, symmetry refers to invariance, the idea that certain properties of a physical system remain unchanged under specific transformations, such as translations or rotations. This concept is not merely aesthetic; it has profound implications for the formulation of physical laws and the conservation principles they entail. For example, the invariance of physical laws under time translation leads to the conservation of energy, while rotational symmetry leads to conservation of angular momentum. As such, symmetry serves not only as a tool for understanding existing phenomena but also as a guiding principle in the pursuit of new physical theories.

In the context of quantum mechanics, one of the most profound consequences of symmetry is encapsulated in Wigner's theorem, including non-bijective isometry variants, which imposes strict constraints on the types of transformations that can be applied to quantum states. Specifically, the theorem asserts that any symmetry transformation preserving transition probabilities between quantum states must be induced by either a unitary or an antiunitary isometry on the system's Hilbert space. This result not only ensures the mathematical consistency of quantum theory under symmetry operations but also reveals how deeply symmetries are woven into the fabric of quantum mechanics, shaping everything from conservation laws to the subtleties of quantum measurement and the computation of event probabilities.

Recent developments in quantum many-body and field theories suggest the existence of symmetries that lie beyond the conventional framework of group theory,

where each symmetry operation has an inverse. These non-invertible symmetries also known as generalized or categorical symmetries lack unique inverses and are better described using topological or higher-categorical structures. They arise naturally in two-dimensional conformal field theories, topological quantum field theories, and certain lattice models, and are closely linked to phenomena such as duality transformations. This broader notion of symmetry hints at a deeper and more intricate structure underlying quantum systems. At first glance, however, non-invertible symmetries appear to stand in tension with Wigner's theorem. The aim of this paper is to resolve this apparent contradiction and propose a generalized extension of Wigner's theorem that accommodates such symmetries.

Ising Model: It is all about Boundary Conditions. We begin by illustrating the issue through well-known examples. A paradigmatic case is the transverse-field Ising chain, which encapsulates the essential features necessary to understand the apparent contradiction. At the outset, it is important to emphasize that the origin of such paradoxes arises from the specific boundary conditions to which quantum systems are subjected.

Consider the transverse-field Ising chain with L sites at the self-dual point associated with two different boundary conditions:

H one equals negative sigma o j zero three plus one minus two o f, j equals one j equals one

H two equals H one minus sigma i minus eta sigma i s i, (one)

where o j equals Pauli matrices acting on site j, and eta hat equals product from j equals one to L sigma j x (two)

is a Z two symmetry common to both Hamiltonians. The first model H one describes a transverse-field Ising chain with open boundary conditions while the closed transverse-field Ising chain H two has a boundary term whose form depends on the eigenvalue of the symmetry operator eta hat. What does it mean for eta hat to be a symmetry? At the risk of sounding overly pedantic, this means that (a) eta hat is a conserved quantity, i.e., commutes with the Hamiltonian, and (b) the action of eta hat must preserve the transition probability between arbitrary quantum states. The latter requirement, which will become essential in what follows, is automatically satisfied if the operator is unitary or antiunitary - the contents of Wigner's original theorem.

Do these Hamiltonians exhibit additional symmetries? As demonstrated in Reference, both H one and H two possess a self-duality transformation (a linear automorphism of bond algebras) implemented by unitary maps whose action on an operator O hat by Phi of O hat minus equals U O hat U dagger, with U unitary. Indeed, the unitary (and Hermitian) operator

U to the power of one equals product from j equals one to floor of L divided by two S j, L minus j plus one product from j equals one to L minus one C j plus one, j x H tensor L, (three)

constructed entirely from elements of the Clifford group, including Controlled-o

C i, j x equals e to the power of i pi divided by four sigma i z sigma j x minus sigma i z minus sigma j x plus one, (four)

Swap, S sub i comma j equals C sub i comma j superscript x C sub j comma i superscript x C sub i comma j superscript x equals one plus sigma sub i dot sigma sub j over two. Which interchanges the spin at site i with the one at site j, and Hadamard

H sub j equals i e to the power of negative i pi over two square roots of two left parenthesis sigma sub j superscript z plus sigma sub j superscript x right parenthesis,

realizes the self-duality map Phi sub one left parenthesis sigma sub j superscript x right parenthesis equals sigma sub r sub j superscript z sigma sub r sub j plus one superscript z z z Phi sub one left parenthesis sigma sub j superscript z sigma sub j plus one superscript z right parenthesis equals sigma sub r sub j superscript x, Phi sub one left parenthesis widehat eta right parenthesis equals sigma sub L superscript z, with r sub j equals L minus j and j equals check one comma cdots comma L minus one, and commutes with H sub one. Similarly, the unitary operator sigma

U superscript left parenthesis two right parenthesis equals left parenthesis product sub j equals one comma L minus one e to the power of negative i pi over four sigma sub j superscript x e to the power of negative i pi over four sigma sub j superscript z sigma sub j plus one superscript z right parenthesis e to the power of negative i pi over four sigma sub L superscript x equals e to the power of negative i pi over four sigma sub one superscript x e to the power of negative i pi over four sigma sub one superscript z sigma sub two superscript z cdots e to the power of negative i pi over four sigma sub L superscript x,

generates the map Phi sub two left parenthesis sigma sub j superscript x right parenthesis equals sigma sub j superscript z sigma sub j plus one superscript z, Phi sub two left parenthesis sigma sub j superscript z sigma sub j plus one superscript z right parenthesis equals sigma sub j plus one superscript x, with j equals one comma cdots comma bar L minus one and Phi sub two left parenthesis sigma sub L superscript x right parenthesis equals widehat eta sigma sub L superscript z sigma sub one superscript z, Phi sub two left parenthesis widehat eta sigma sub L superscript z sigma sub one superscript z right parenthesis equals sigma sub one superscript x, Phi sub two left parenthesis widehat eta right parenthesis equals widehat eta, and commutes with the Hamiltonian H two. Consequently, as argued in references, these two unitary transformations qualify as respective symmetries of the system when the corresponding boundary conditions are implemented. We note that deriving explicit expressions for the unitary operators implementing the dualities is a notably challenging task. For example, even given U superscript left parenthesis two right parenthesis, deducing the form of U superscript left parenthesis one right parenthesis is nontrivial and, to the best of our knowledge, has not appeared in the literature. The operator encodes the automorphism's entanglement characteristics and depends sensitively on the boundary conditions.

Consider now the same TFIC, but with explicit periodic or antiperiodic boundary condition:

H superscript plus minus equals H sub one minus sigma sub L superscript x plus minus sigma sub L superscript z sigma sub one superscript z.

The operators U superscript left parenthesis one comma two right parenthesis are no longer conserved quantities of H superscript plus minus. In this case, no automorphism exists; instead one may attempt to define non-invertible operators

D sub plus equals U superscript two P sub plus, (eight)

that, respectively, commute with the Hamiltonians H superscript plus of Equation seven, i.e., H superscript plus and D sub plus equals zero. Here, P sub plus equals one plus eta hat divided by two denote projection operators P sub plus squared equals P sub plus onto the positive/negative symmetry eigenvalue eta equals plus or minus one sectors. Importantly, however, the transition probability between two arbitrary quantum states ket alpha, ket beta in the Hilbert space script H of dimension two superscript L is no longer conserved after the action by D sub plus. Indeed, explicitly implementing such a transformation,

(nine)

ket bra D sub plus beta bra D sub plus alpha ket squared equals ket bra beta P sub plus U superscript two dagger U superscript two P sub plus alpha ket squared equals ket bra beta P sub plus alpha ket squared not equal to ket bra beta alpha ket squared. In other words, despite being a conserved quantity, D sub plus does not qualify as a symmetry according to Wigner, since its probability altering action can be detected by measurements-i.e., the application of D sub plus text does not lead to an invariance of the system. This example thus underscores that even modest changes to the boundary conditions may carry radical consequences for the nature (and existence) of the system automorphisms. As we will detail towards the end of this paper, bona fide symmetry operators may, nonetheless, be explicitly written down for an amended (gauged) TFIC.

Generalized Symmetry Extension of Wigner's Theory. A key assumption in Wigner's theorem is that symmetry operators act on the same Hilbert space in which quantum states are defined; consequently, all such symmetries are inherently invertible. Since the preservation of transition probabilities is the central physical principle, we propose an extension of Wigner's theorem to include non-invertible (symmetry) operators.

Our extension of Wigner's theorem to non-invertible symmetries asserts that: Any non-invertible symmetry transformation that preserves transition probabilities between quantum states of a Hilbert space, can only be induced by the composition of either a unitary or an antiunitary operator and a projector onto a symmetry sector of an enlarged Hilbert space - such that this projection operator acts as the identity on the original Hilbert space. Our proof applies to Hilbert spaces of finite or countably infinite (denumerably infinite) dimension.

Preliminaries:

Consider a quantum system with a Hermitian Hamiltonian H having its support in a Hilbert space H defined over the field of the complex numbers C. The dimension of H, denoted dimension H, is assumed to be either finite or denumerably infinite. Any pair of arbitrary normalized states alpha, beta in H can be expanded in the orthonormal eigenbasis of H, i.e., H lambda alpha equals E sub p lambda alpha (E sub p in R and alpha sub mu nu).

alpha ket equals sum aplu alpha ket and beta ket equals sum bp beta ket,

with alpha sub u equals ap sub a, beta sub p equals ap sub beta in C. The above system can, in general, be extended to a larger (finite or denumerably infinite of dimension dimension H greater than dimension H) space H equals H plus H sub one, with normalized states alpha sub one in H sub one, a subspace orthogonal to H. The subsystem H sub one in the tensor product extension is spanned by orthonormal states ket gp. On this larger space H, the above Hamiltonian H is replaced by a "gauge-enlarged" Hamiltonian H sub G; the original Hamiltonian H may be understood as H equals P sub H H sub G P sub H, where P sub H is a projection operator onto H in H.

We now return to our central focus: the investigation of non-invertible symmetries. Suppose there exists a conserved quantity D, defined on H, i.e.,

H and D equals zero, (eleven)

and D is represented by a bounded but non-invertible operator. The corresponding operator D, defined on the extended space H, satisfies

D equals P sub H D P sub H, (twelve)

meaning that the action of the H-component of D on states in H coincides exactly with the action of D on those states. The action of D on the pair of states alpha ket and beta ket in H is denoted by alpha ket equals D alpha, beta ket equals D beta. (thirteen)

We stress that D need not leave H invariant and thus alpha ket and beta ket may, generally, lie outside H. Recall that automorphisms may be linear or antilinear; antilinear maps can be written as the composition of a linear map with complex conjugation K.

Thus, for the associated transition probability - defined via the inner product structure of the Hilbert space - to remain invariant, it must hold that for all such arbitrary states alpha ket, beta ket in H, the conditions

K alpha ket squared equals beta ket squared equals beta D alpha squared equals beta alpha squared, (fourteen)

and, for all alpha sub one ket in H sub one,

K alpha sub one ket alpha squared equals D alpha sub one D alpha squared equals alpha sub one alpha squared equals zero, (fifteen)

must be satisfied.