# Bosonic and fermionic Gaussian states from Kähler structures

### Submission summary

 As Contributors: Lucas Hackl Arxiv Link: https://arxiv.org/abs/2010.15518v2 (pdf) Date submitted: 2020-11-30 04:55 Submitted by: Hackl, Lucas Submitted to: SciPost Physics Academic field: Physics Specialties: Mathematical Physics Condensed Matter Physics - Theory Quantum Physics Statistical and Soft Matter Physics Approach: Theoretical

### Abstract

We show that bosonic and fermionic Gaussian states (also known as "squeezed coherent states") can be uniquely characterized by their linear complex structure $J$ which is a linear map on the classical phase space. This extends conventional Gaussian methods based on covariance matrices and provides a unified framework to treat bosons and fermions simultaneously. Pure Gaussian states can be identified with the triple $(G,\Omega,J)$ of compatible K\"ahler structures, consisting of a positive definite metric $G$, a symplectic form $\Omega$ and a linear complex structure $J$ with $J^2=-1\!\!1$. Mixed Gaussian states can also be identified with such a triple, but with $J^2\neq -1\!\!1$. We apply these methods to show how computations involving Gaussian states can be reduced to algebraic operations of these objects, leading to many known and some unknown identities. From this, we compiled a comprehensive list of mathematical structures and formulas to compare bosonic and fermionic Gaussian states side-by-side.

###### Current status:
Editor-in-charge assigned

### Submission & Refereeing History

Submission 2010.15518v2 on 30 November 2020

## Reports on this Submission

### Strengths

1. The authors work out in detail the general case of canonical quantized systems that can be exactly solved: arbitrary quadratic hamiltonians and complex structures, for arbitrary bosonic and fermionic degrees of freedom. This provides a valuable reference and starting point for a wide array of research directions, some of which are described in the final section.

2. The approach to the material brings together a structural, coordinate-invariant view, together with detailed formulas in coordinates, making clear how things depend on coordinate choices. Other references typically are abstract, leaving it unclear how to do calculations, or just a collection of complicated formulas, leaving their structure unclear.

3. The approach is exhaustive, working out everything in detail, providing formulas useful in a wide variety of contexts, not available elsewhere.

4. The writing is unusually clear, easy to follow and very good at providing insight into what otherwise can be an obscure and difficult to follow formalism.

### Weaknesses

1. Most of what the authors are doing follows from basic ideas that have been known to experts for a long time.
Much of what they write down that is not available elsewhere could be worked out with a moderate amount of effort by a researcher who needed the results.

### Report

I recommend publication, this is a quite valuable document. It meets the journal criteria by providing a basis for future research, linking a wide variety of different research areas.

### Requested changes

No changes needed.

• validity: top
• significance: high
• originality: high
• clarity: top
• formatting: perfect
• grammar: perfect

### Report

See report

• validity: -
• significance: -
• originality: -
• clarity: -
• formatting: -
• grammar: -