The Gauge Picture of Quantum Dynamics

2025-05-06 0 0 971.42KB 11 页 10玖币

Kevin Slagle1,2,3

1Department of Electrical and Computer Engineering, Rice University, Houston, Texas 77005 USA

2Department of Physics, California Institute of Technology, Pasadena, California 91125, USA

3Institute for Quantum Information and Matter and Walter Burke Institute for Theoretical Physics, California Institute of Technology,

Pasadena, California 91125, USA

Although local Hamiltonians exhibit local

time dynamics, this locality is not explicit

in the Schr¨

odinger picture in the sense that

the wavefunction amplitudes do not obey a

local equation of motion. We show that

geometric locality can be achieved explicitly

in the equations of motion by “gauging”

the global unitary invariance of quantum

mechanics into a local gauge invariance. That

is, expectation values ⟨ψ|A|ψ⟩are invariant

under a global unitary transformation acting

on the wavefunction |ψ⟩ → U|ψ⟩and operators

A→UAU†, and we show that it is possible

to gauge this global invariance into a local

gauge invariance. To do this, we replace

the wavefunction with a collection of local

wavefunctions |ψJ⟩, one for each patch of space

J. The collection of spatial patches is chosen

to cover the space; e.g. we could choose the

patches to be single qubits or nearest-neighbor

sites on a lattice. Local wavefunctions

associated with neighboring pairs of spatial

patches Iand Jare related to each other by

dynamical unitary transformations UIJ . The

local wavefunctions are local in the sense

that their dynamics are local. That is, the

equations of motion for the local wavefunctions

|ψJ⟩and connections UIJ are explicitly local in

space and only depend on nearby Hamiltonian

terms. (The local wavefunctions are many-

body wavefunctions and have the same Hilbert

space dimension as the usual wavefunction.)

We call this picture of quantum dynamics

the gauge picture since it exhibits a local

gauge invariance. The local dynamics of a

single spatial patch is related to the interaction

picture, where the interaction Hamiltonian

consists of only nearby Hamiltonian terms.

We can also generalize the explicit locality

to include locality in local charge and energy

densities.

Contents

1 Introduction 2

2 Schr¨odinger to Heisenberg Picture 2

2.1 Locality ................. 3

3 Gauge Picture 3

3.1 Solving the Equations of Motion . . . 5

3.2 Generalized Hamiltonians . . . . . . . 5

4 Local Interaction Picture 6

5 Quantum Circuits 6

5.1 Unitary Operators . . . . . . . . . . . 6

6 Measurements 7

7 Generalized Locality 9

8 Conclusion 9

8.1 New Approximation Technique . . . . 10

8.2 New Deformation of Quantum Mechanics 10

8.3 Other Future Directions . . . . . . . . 10

Acknowledgments 10

References 11

Accepted in Quantum 2024-03-13, click title to verify. Published under CC-BY 4.0. 1

arXiv:2210.09314v5 [quant-ph] 15 Mar 2024

1 Introduction

Locality is of fundamental importance in theoretical

physics. The observable physics of quantum dynamics

is local if the Hamiltonian is geometrically local1,

i.e. if the Hamiltonian is a sum of local operators.

An operator (other than the Hamiltonian) is said to

be local if it only acts on a small region of space.

But locality is not explicit in the Schr¨odinger picture

because the wavefunction is global in the sense that it

can not be associated with any local region of space,

and the time dynamics of the wavefunction globally

depends on all Hamiltonian terms. On the other

hand, locality is explicit in the Heisenberg picture,

for which the time dynamics of local operators

only depends on nearby local operators (when the

Hamiltonian is local). [1] This motivates us to ask: Is

it possible to modify the Schr¨odinger picture to make

locality explicit in the equations of motion?

Gauge theory is another fundamental concept in

theoretical physics. Gauge theory is the foundation

of the Standard Model of particle physics and is also

used to describe exotic phases of condensed matter

[2,3]. An important tool that gauge theory provides

is the gauging process, in which one promotes a

global symmetry into a local gauge symmetry by

coupling the original model to gauge ﬁelds. For

example, a scalar ﬁeld theory L=1

2(∂µϕ)2is invariant

under a global U(1) symmetry ϕ(x)→ϕ(x) + λ.

By coupling ϕto a gauge ﬁeld Aµas in

Lgauged =1

2(∂µϕ−Aµ)2, the global symmetry

is promoted to a local gauge symmetry where

ϕ(x)→ϕ(x) + λ(x)and Aµ(x)→Aµ(x) + ∂µλ(x).

Gauging more exotic symmetries leads to more

exotic physics; e.g. gauging spatial symmetries can

lead to gravity and gauging fractal symmetries can

result in fracton topological order [4]. In quantum

mechanics, expectation values ⟨ψ|A|ψ⟩are invariant

under a global unitary transformation acting on the

wavefunction |ψ⟩ → U|ψ⟩and operators A→UAU†.

Although this transformation is typically viewed as a

global invariance rather than a global symmetry, we

can still ask: Is it possible to gauge the global unitary

invariance in quantum mechanics? And what are the

consequences of doing so?

We ﬁnd that the answer to both questions is

yes, and that one consequence of gauging the global

unitary invariance is that locality becomes explicit

in the equations of motion. In order to achieve

this, we introduce a collection of local wavefunctions

|ψI⟩, each associated with a local patch of space.

Each local wavefunction is an element of the same

Hilbert space as the usual wavefunction. The local

wavefunction associated with nearby patches are

related by unitary transformations UIJ , which are

1Geometric locality is not to be confused with the weaker

notion of k-locality, for which an operator is k-local if it acts

on at most kqubits anywhere in space.

also dynamical. Locality is explicit in the sense that

the equations of motion [Eq. (19)] for the dynamical

variables (|ψI⟩and UIJ ) only depends on nearby

Hamiltonian terms and nearby dynamical variables.

(Note that locality is not explicit in Schr¨odinger’s

picture in this way.)

The Schr¨odinger and Heisenberg pictures are

related by a time-dependant unitary transformation

that acts on the wavefunctions and operators. Our

new picture of quantum dynamics is related to the

Schr¨odinger and Heisenberg pictures via a local gauge

transformation. We describe these derivations in

detail in Sec. 2and Sec. 3. The new local equations

of motion are given in Eq. (19), and the local gauge

transformation is given in Eq. (22).

In Sec. 4, we show that the gauge picture local

wavefunction associated with a patch is equivalent

to the interaction picture wavefunction when the

interaction Hamiltonian is the sum over Hamiltonian

terms that have some support on that patch. To

gain intuition, in Sec. 5we consider the example

of a quantum circuit in the gauge picture. In

Sec. 6, we describe the measurement process in the

gauge picture. Although local unitary dynamics are

explicitly local in the gauge picture, the aﬀect of

local measurement is not explicitly local in the gauge

picture (similar to the Schr¨odinger picture). In Sec. 7,

we generalize spatial locality in the gauge picture to

e.g. locality in local particle number or local energy

density.

2 Schr¨

odinger to Heisenberg Picture

To warm up, we ﬁrst derive the Heisenberg picture

from the Schr¨odinger picture. In the Schr¨odinger

picture, the wavefunction evolves according to

∂tψS(t)=−iHS(t)ψS(t)(1)

where HS(t)is the Hamiltonian (and we set ℏ= 1).

We use superscripts “S” and “H” to respectively

label time-dependent variables in the Schr¨odinger and

Heisenberg pictures.

To derive the Heisenberg picture, consider applying

a time-dependent unitary transformation to the

wavefunction and operators:

ψH(t)=U†(t)ψS(t)

AH(t) = U†(t)AS(t)U(t)(2)

Equation (2)deﬁnes the meaning of the “H”

superscript for all operators and wavefunctions. This

unitary transformation has the essential property that

it does not aﬀect expectation values:

ψH(t)AH(t)ψH(t)=ψS(t)AS(t)ψS(t)(3)

Since U(t)does not aﬀect the physics, the unitary

transformation U(t)could therefore be viewed as a

Accepted in Quantum 2024-03-13, click title to verify. Published under CC-BY 4.0. 2

global “gauge” transformation, which we will use to

move the time dynamics from the wavefunction to the

operators.

Let GS(t)be the Hermitian operator such that

∂tU(t) = −iGS(t)U(t) (4)

Or equivalently, ∂tU(t) = −iU(t)GH(t)where

GH(t) = U†(t)GS(t)U(t)is deﬁned by Eq. (2).

The time derivatives of the new wavefunction and

operators are

∂tψH(t)=−iHH(t)ψH(t)+iGH(t)ψH(t)

∂tAH(t) = (∂tAS)H(t) + i[GH(t), AH(t)]

(5)

where (∂tAS)H(t) = U†(t)∂tAS(t)U(t)[Eq. (2)].

To obtain the Heisenberg picture, we simply choose

GH(t) = HH(t) (6)

with the initial condition U(0) = 1at t= 0,

where 1denotes the identity operator. This choice

makes the wavefunction |ψH⟩constant in time, while

operators evolve in time. If the Hamiltonian is

time-independent, then the Hamiltonian is the same

in the Schr¨odinger and Heisenberg pictures; i.e.

HS(t) = HH(t)if ∂tHS= 0.

2.1 Locality

A nice feature of the Heisenberg picture is that if

the Hamiltonian is local, then the time evolution of

observables is explicitly local. A local Hamiltonian

has the form

H=X

HJ(7)

where each HJonly acts on a ﬁnite patch of space J.

We use capital letters, I,J, and K, to denote patches

of space.

Now consider the time evolution of a local operator

in the Heisenberg picture:

∂tAH

I(t) = i[HH(t), AH

I(t)] + (∂tAS

I)H(t) (8)

Throughout this work, AIdenotes a local operator

that acts within the spatial patch I(when viewed in

the Schr¨odinger picture), and similar for BJ, etc. Due

to locality, most Hamiltonian terms cancel out in the

ﬁrst term. The result is an explicitly local Heisenberg

equation of motion:

∂tAH

I(t) = i[HH

⟨I⟩(t), AH

I(t)] + (∂tAS

I)H(t) (9)

where (∂tAS

I)H(t) = U†(t)(∂tAS

I)S(t)U(t)[Eq. (2)].

H⟨I⟩is a sum over nearby Hamiltonian terms:

H⟨I⟩=J∩I̸=∅

HJ(10)

ψIψJ

Figure 1: An example of a chain of qubits (black dots)

and spatial patches (colored ovals) consisting of pairs of

neighboring qubits. A local wavefunction |ψI⟩is associated

with each patch I, and the Hilbert spaces associated with

neighboring patches are related by unitary connections UIJ .

PJ∩I̸=∅

Jdenotes a sum over patches Jthat have

overlap with patch I. Note that the local Hamiltonian

terms HH

I(t)also evolve according to Eq. (9).

Locality is explicit in this local Heisenberg picture

because the time evolution of each local operator

I(t)only depends on nearby time-dependent

operators HH

J(t). Locality is not explicit in

Schr¨odinger’s picture since the time evolution of the

wavefunction globally depends on all Hamiltonian

terms, and there is no sense in which the wavefunction

(or parts of it) can be associated with a point in space.

3 Gauge Picture

We now want to obtain a local picture of quantum

dynamics that features time-dependent wavefunctions

and time-independent operators. To do this, we

ﬁrst choose a set of local patches of space that

cover the entire space (or lattice); see Fig. 1for an

example. Then for each patch of space, we apply a

time-dependent unitary transformation UI(t)to the

Heisenberg picture wavefunction and local operators:

ψI=UIψH

I=UIAH

IU†

(11)

We thus obtain a separate local wavefunction |ψI⟩for

each patch of space. For a certain transformation

UI, the wavefunctions |ψI⟩are local in the sense

that their dynamics are local and only depend on

nearby Hamiltonian terms. Note that |ψI⟩is a many-

body wavefunction that belongs to the same Hilbert

space as the usual wavefunction. The second line

transforms local operators (including HIand H⟨I⟩)

from the Heisenberg picture into a new picture of

quantum dynamics. The “G” superscript labels time-

dependent operators that evolve within this picture.

We omit the “G” superscript for |ψI⟩since the |ψI⟩

notation is not used in other pictures; thus, there

should be no confusion. To further reduce clutter, we

also suppress the “(t)” notation for time-dependent

variables.

After applying this unitary transformation, cor-

relation functions within a single patch become

⟨ψH|AH

I|ψH⟩=⟨ψI|AG

I|ψI⟩. Correlation functions of

Accepted in Quantum 2024-03-13, click title to verify. Published under CC-BY 4.0. 3

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TheGaugePictureofQuantumDynamicsKevinSlagle1,2,31DepartmentofElectricalandComputerEngineering,RiceUniversity,Houston,Texas77005USA2DepartmentofPhysics,CaliforniaInstituteofTechnology,Pasadena,California91125,USA3InstituteforQuantumInformationandMatterandWalterBurkeInstituteforTheoreticalPhysics,Cali...

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The Gauge Picture of Quantum Dynamics

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