This paper develops an analogy proposed by Stachel between general relativity (GR) and quantum mechanics (QM) as regards permutation invariance. Our main idea is to overcome Pooley's criticism of the analogy by appeal to paraparticles. In GR the equations are (the solution space is) invariant under diffeomorphisms permuting spacetime points. Similarly, in QM the equations are invariant under particle permutations. Stachel argued that this feature--a theory's `not caring which point, or particle, is which'--supported a structuralist ontology. Pooley criticizes this analogy: in QM the (anti-)symmetrization of fermions and bosons implies that each individual state (solution) is fixed by each permutation, while in GR a diffeomorphism yields in general a distinct, albeit isomorphic, solution. We define various versions of structuralism, and go on to formulate Stachel's and Pooley's positions, admittedly in our own terms. We then reply to Pooley. Though he is right about fermions and bosons, QM equally allows more general types of symmetry, in which states (vectors, rays or density operators) are not fixed by all permutations (called `paraparticle states'). Thus Stachel's analogy is revived.This paper develops an analogy proposed by Stachel between general relativity (GR) and quantum mechanics (QM) as regards permutation invariance. Our main idea is to overcome Pooley's criticism of the analogy by appeal to paraparticles. In GR the equations are (the solution space is) invariant under diffeomorphisms permuting spacetime points. Similarly, in QM the equations are invariant under particle permutations. Stachel argued that this feature--a theory's `not caring which point, or particle, is which'--supported a structuralist ontology. Pooley criticizes this analogy: in QM the (anti-)symmetrization of fermions and bosons implies that each individual state (solution) is fixed by each permutation, while in GR a diffeomorphism yields in general a distinct, albeit isomorphic, solution. We define various versions of structuralism, and go on to formulate Stachel's and Pooley's positions, admittedly in our own terms. We then reply to Pooley. Though he is right about fermions and bosons, QM equally allows more general types of symmetry, in which states (vectors, rays or density operators) are not fixed by all permutations (called `paraparticle states'). Thus Stachel's analogy is revived.

arXiv:1002.3730Feb 19, 20100 citations

On the description of fermion systems in boson representations

Bent E. Sørensen

Nuclear Physics AApr 1, 1967105 citations

Duality and bosonization of (2+1) -dimensional Majorana fermions

Max A. Metlitski, Ashvin Vishwanath, Cenke Xu

We construct a dual-bosonized description of a massless Majorana fermion in (2+1) $d$ . In contrast to Dirac fermions, for which a bosonized description can be constructed using a flux attachment procedure, neutral Majorana fermions call for a different approach.We argue that the dual theory is an $\mathrm{SO}{(N)}_{1}$ Chern-Simons gauge theory with a critical $\mathrm{SO}(N)$ vector bosonic matter field ($N\ensuremath{\ge}3$). The monopole of the $\mathrm{SO}(N)$ gauge field is identified with the Majorana fermion. We provide evidence for the duality by establishing the correspondence of adjacent gapped phases and by a parton construction. We also propose a generalization of the duality to ${N}_{f}$ flavors of Majorana fermions, and discuss possible resolutions of a caveat associated with an emergent global ${Z}_{2}$ symmetry. Finally, we conjecture a dual description of an $\mathcal{N}=1$ supersymmetric fixed point in (2+1) $d$ , which is realized by tuning a single flavor of Majorana fermions to an interacting (Gross-Neveu) critical point.We construct a dual-bosonized description of a massless Majorana fermion in (2+1)$d$. In contrast to Dirac fermions, for which a bosonized description can be constructed using a flux attachment procedure, neutral Majorana fermions call for a different approach.We argue that the dual theory is an $\mathrm{SO}{(N)}_{1}$ Chern-Simons gauge theory with a critical $\mathrm{SO}(N)$ vector bosonic matter field ($N\ensuremath{\ge}3$). The monopole of the $\mathrm{SO}(N)$ gauge field is identified with the Majorana fermion. We provide evidence for the duality by establishing the correspondence of adjacent gapped phases and by a parton construction. We also propose a generalization of the duality to ${N}_{f}$ flavors of Majorana fermions, and discuss possible resolutions of a caveat associated with an emergent global ${Z}_{2}$ symmetry. Finally, we conjecture a dual description of an $\mathcal{N}=1$ supersymmetric fixed point in (2+1)$d$, which is realized by tuning a single flavor of Majorana fermions to an interacting (Gross-Neveu) critical point.

arXiv:1611.05049Physical review. B./Physical review. BMay 23, 201758 citations

We construct a dual-bosonized description of a massless Majorana fermion in (2+1)

d

. In contrast to Dirac fermions, for which a bosonized description can be constructed using a flux attachment procedure, neutral Majorana fermions call for a different approach.We argue that the dual theory is an

\mathrm{SO}{(N)}_{1}

Chern-Simons gauge theory with a critical

\mathrm{SO}(N)

vector bosonic matter field ($N\ensuremath{\ge}3$). The monopole of the

\mathrm{SO}(N)

gauge field is identified with the Majorana fermion. We provide evidence for the duality by establishing the correspondence of adjacent gapped phases and by a parton construction. We also propose a generalization of the duality to

{N}_{f}

flavors of Majorana fermions, and discuss possible resolutions of a caveat associated with an emergent global

{Z}_{2}

symmetry. Finally, we conjecture a dual description of an

\mathcal{N}=1

supersymmetric fixed point in (2+1)

d

, which is realized by tuning a single flavor of Majorana fermions to an interacting (Gross-Neveu) critical point.