Localisation problem for relativistic detectors in the Unruh-DeWitt model

Evan Galea, and Magdalena Zycha, b

aARC Centre of Excellence for Engineered Quantum Systems, School of Mathematics and Physics,
The University of Queensland, St Lucia, QLD 4072, Australia
bDepartment of Physics, Stockholm University, SE-106 91 Stockholm, Sweden
 

Physical Review D 107, 056023 (2023)

  
  

Relativistic quantum physics

  • 1900s: Einsteinian revolution—special relativity and quantum mechanics
    •  
  • Mid-1920s: theories unified by relativistic quantum field theory
    •  
  • Still many open questions:
    • How should one define a 'particle' in quantum theory?
    • What can the transition from relativistic quantum field theory to non-relativistic quantum mechanics tell us?
    • Can one define a position operator in relativistic quantum theory?

Unruh-DeWitt (UDW) detector model

Originally proposed by Unruh [1], then simplified by DeWitt [2].

Couples a two-level system to a quantised scalar field $\hat{\phi}(\vec{x})$

\[ \hat{H}_I = \lambda \hat{\mu} \otimes \hat{\phi}(\vec{x}) \]

Polarisation or 'monopole' operator exchanges energy levels

\[ \hat{\mu} = | g \rangle \langle e | + | e \rangle \langle g | \]
[1] W. G. Unruh, Phys. Rev. D 14, 870 (1976).
[2] B. DeWitt, in General Relativity: An Einstein Centenary Survey (University Press, UK, 1979).

Unruh-DeWitt (UDW) detector model

Originally proposed by Unruh [1], then simplified by DeWitt [2].

Couples a two-level system to a quantised scalar field $\hat{\phi}(\vec{x})$

\[ \hat{H}_I = \lambda \hat{\mu} \otimes \hat{\phi}(\vec{x}) \]

Field can be decomposed as superposition of plane wave modes

\[ \hat{\phi}(\vec{x}) = \frac{1}{(2\pi)^{3/2}} \int \frac{d^3k}{\sqrt{2 \omega(\vec{k})}} \Big ( \hat{a}(\vec{k}) e^{-i \vec{k} \cdot \vec{x}} + \mathrm{H.c.} \Big ) \]
[1] W. G. Unruh, Phys. Rev. D 14, 870 (1976).
[2] B. DeWitt, in General Relativity: An Einstein Centenary Survey (University Press, UK, 1979).

Detectors with quantised centre of mass

Considered by Unruh and Wald for a box detector [3]

\[ \hat{H}_I = \lambda \int d^3x \, \hat{\mu}(\vec{x}) \otimes \hat{\phi}(\vec{x}) \]

with a position-dependent monopole operator of the form

\[ \hat{\mu}(\vec{x}) \equiv \psi_g^*(\vec{x}) \psi_e(\vec{x}) | g \rangle \langle e | + \psi_e^*(\vec{x}) \psi_g(\vec{x}) | e \rangle \langle g | \]

[3] W. G. Unruh and R. M. Wald, Phys. Rev. D 29, 1047 (1984).

Detectors with quantised centre of mass

Alternative extension considered by Stritzelberger and Kempf [4]

\[ \hat{H}_I = \lambda \int d^3x \, \hat{\mu} \otimes | \vec{x} \rangle \langle \vec{x} |_D \otimes \hat{\phi}(\vec{x}) \]

Detector's free Hamiltonian now includes a kinetic term

\[ \hat{H}_D = \frac{\hat{\vec{p}}^2}{2 M} + E | e \rangle \langle e | \]

Full atom-light interaction treated by Wilkens [5]


[4] N. Stritzelberger and A. Kempf, Phys. Rev. D 101, 036007 (2020).
[5] M. Wilkens, Phys. Rev. A 47, 671 (1993).

Relativistic centre of mass?

Problem: both extensions assume a non-relativistic detector!


  • Model mixes Lorentz and Galilean groups
    •  
  • Leads to "spurious friction forces" acting on detector [5]

Relativistic centre of mass?

Problem: both extensions assume a non-relativistic detector!


Can introduce relativistic centre of mass in two different ways:

  • Second-quantised model—relativistic quantum field theory
    •  
  • First-quantised model—relativistic quantum mechanics

Let's consider both models, starting with second-quantised case

Relativistic second-quantised detector

Relativistic second-quantised detector model proposed by Unruh [1]

  • Replace energy levels of detector by fields $| j \rangle \to \hat{\psi}_j$
    • \[ \hat{\mu}_\hat{\psi}(\vec{x}) = \hat{\psi}^\dagger_g(\vec{x}) \hat{\psi}_e(\vec{x}) + \hat{\psi}^\dagger_e(\vec{x}) \hat{\psi}_g(\vec{x}) \]
  • The interaction Hamiltonian takes the form
    • \[ \hat{H}_I^{(\mathrm{2nd})} = \lambda^{(\mathrm{2nd})} \int d^3x \, \hat{\mu}_\hat{\psi}(\vec{x}) \otimes \hat{\phi}(\vec{x}) \]
  • Coupling constants between models are dimensionally distinct!

Relativistic second-quantised detector

Giacomini and Kempf [6] simplified by restricting to one-particle sector

  • Interaction couples position and internal degrees of freedom
    • \[ \hat{H}_I^{(\mathrm{2nd})} \Big |_{\mathcal{H}^D_1} = \lambda^{(\mathrm{2nd})} \int d^3x \, \Big( | \vec{x}_g \rangle \langle \vec{x}_e |_D^{(\mathrm{2nd})} + | \vec{x}_e \rangle \langle \vec{x}_g |_D^{(\mathrm{2nd})} \Big) \otimes \hat{\phi}(\vec{x}) \]
  • Looks similar to previous non-relativistic model, except that
    • \[ | \vec{x}_j \rangle_D^{(\mathrm{2nd})} \equiv \frac{1}{(2\pi)^{3/2}} \int \frac{d^3p}{\sqrt{2 E_j(\vec{p})}} e^{-i \vec{p} \cdot \vec{x}} | \vec{p}, j \rangle_D \]
  • Position states are not Fourier transforms of momentum eigenstates!

[6] F. Giacomini and A. Kempf, Phys. Rev. D 105, 125001 (2022).

A relativistic first-quantised model?

Quantise the detector in a boosted frame with momentum $\vec{p}$

\[ \hat{H}_D = \sqrt{\hat{\vec{p}}^2 + \hat{M}^2} \, . \]

To describe internal energy, we must quantise the mass-energy

\[ \hat{M} = M_g | g \rangle \langle g | + M_e | e \rangle \langle e | \, , \]

where

\[ \hat{M} | j \rangle = M_j | j \rangle \, . \]

A relativistic first-quantised model?

Interaction Hamiltonian should have the form

\[ \hat{H}_I^{(\mathrm{1st})} = \lambda^{(\mathrm{1st})} \int d^3x \, \hat{\mu} \otimes | \vec{x} \rangle \langle \vec{x} |_D^{(\mathrm{1st})} \otimes \hat{\phi}(\vec{x}) \, . \]

This Hamiltonian coincides with the non-relativistic expression

A relativistic first-quantised model?

Interaction Hamiltonian should have the form

\[ \hat{H}_I^{(\mathrm{1st})} = \lambda^{(\mathrm{1st})} \int d^3x \, \hat{\mu} \otimes | \vec{x} \rangle \langle \vec{x} |_D^{(\mathrm{1st})} \otimes \hat{\phi}(\vec{x}) \, . \]

But we have to be a little careful!

    Q. What time parameter does the Hamiltonian generate time translations for? [8]

    A. Inertial detector. Generate time translations wrt coordinate time $t$


[8] E. Martín-Martínez and P. Rodriguez-Lopez, Phys. Rev. D 97, 105026 (2018).

A relativistic first-quantised model?

Interaction Hamiltonian should have the form

\[ \hat{H}_I^{(\mathrm{1st})} = \lambda^{(\mathrm{1st})} \int d^3x \, \hat{\mu} \otimes | \vec{x} \rangle \langle \vec{x} |_D^{(\mathrm{1st})} \otimes \hat{\phi}(\vec{x}) \, . \]

But we have to be a little careful!

    Q. How do we define the position states $| \vec{x} \rangle_D^{(\mathrm{1st})}$?

    A. Take states to be Fourier transform of momentum eigenstates

    \[ | \vec{x} \rangle_D^{(\mathrm{1st})} \equiv \frac{1}{(2\pi)^{3/2}} \int d^3p \, e^{-i \vec{p} \cdot \vec{x}} | \vec{p} \rangle_D \, . \]

A relativistic first-quantised model?

Interaction Hamiltonian should have the form

\[ \hat{H}_I^{(\mathrm{1st})} = \lambda^{(\mathrm{1st})} \int d^3x \, \hat{\mu} \otimes | \vec{x} \rangle \langle \vec{x} |_D^{(\mathrm{1st})} \otimes \hat{\phi}(\vec{x}) \, . \]

But we have to be a little careful!

    Q. How are the coupling constants between models related?

    A. Take couplings to coincide for detector at rest

    \[ \lambda^{\mathrm{(2nd)}} = \sqrt{2 (M_g^2 + M_e^2)} \, \lambda^{\mathrm{(1st)}} \, . \]

Summary

First-quantised model

\[ \hat{H}_I^{(\mathrm{1st})} = \lambda^{(\mathrm{1st})} \int d^3x \, \Big( | \vec{x}_g \rangle \langle \vec{x}_e |_D^{(\mathrm{1st})} + | \vec{x}_e \rangle \langle \vec{x}_g |_D^{(\mathrm{1st})} \Big) \otimes \hat{\phi}(\vec{x}) \, . \]

Second-quantised model

\[ \hat{H}_I^{(\mathrm{2nd})} \Big |_{\mathcal{H}^D_1} = \lambda^{(\mathrm{2nd})} \int d^3x \, \Big( | \vec{x}_g \rangle \langle \vec{x}_e |_D^{(\mathrm{2nd})} + | \vec{x}_e \rangle \langle \vec{x}_g |_D^{(\mathrm{2nd})} \Big) \otimes \hat{\phi}(\vec{x}) \, . \]

Different localisations are given by

\[ | \vec{x}_j \rangle_D^{(\mathrm{1st})} \equiv \frac{1}{(2\pi)^{3/2}} \int d^3p \, e^{-i \vec{p} \cdot \vec{x}} | \vec{p}, j \rangle_D \, , \quad | \vec{x}_j \rangle_D^{(\mathrm{2nd})} \equiv \frac{1}{(2\pi)^{3/2}} \int \frac{d^3p}{\sqrt{2 E_j(\vec{p})}} e^{-i \vec{p} \cdot \vec{x}} | \vec{p}, j \rangle_D \, . \]

Transition rate for spontaneous emission

  • We idealise the atom-light interaction, finding transition from
    • \[ | \Psi_i \rangle = | \psi_i, e \rangle_D \otimes | 0 \rangle \, , \text{ with } \, | \psi_i \rangle = \int{d^3p \, \psi_i(\vec{p}; \vec{p}_0) | \vec{p}} \rangle \, , \]

    to the final state via spontaneous emission of the atom-detector

    \[ | \Psi_f \rangle = | \vec{p}_f, g \rangle_D \otimes \hat{a}^\dagger(\vec{k}) | 0 \rangle \, . \]
  • Interaction picture: evolution wrt free Hamiltonian $\hat{H}_D + \hat{H}_F$
  • Weak coupling: find to first-order in perturbation theory
    • \[ \hat{U}(t_f, \, t_i) = 1 - i \int_{t_i}^{t_f} dt \, \hat{H}_{I}(t) + \mathcal{O}(\lambda^2) \, . \]

Transition rate for spontaneous emission

Transition amplitude and probability found from

\[ \mathcal{A}_{| \vec{p}_i, e, 0 \rangle \to | \vec{p}_f, g, 1_{\vec{k}} \rangle} = \langle \Psi_f | \hat{U}(t_f, \, t_i) | \Psi_i \rangle \, , \\ P_{| \vec{p}_i, e, 0 \rangle \to | g \rangle} = \int d^3k \int d^3p_f \, \left| \mathcal{A}_{| \vec{p}_i, e, 0 \rangle \to | \vec{p}_f, g, 1_{\vec{k}} \rangle} \right|^2 \, . \]

Transition rate given as a functional of the detector wave function (probability density function), convolved with a 'template function'

\[ \dot{P}[\psi_i] = \frac{\lambda^2}{2 \pi} \int d^3p \, |\psi_i(\vec{p}; \vec{p}_0)|^2 \, \mathcal{T}_{\mathrm{rel}}(\vec{p}) \, . \]

Transition rate for spontaneous emission

Transition rate given as a functional of the detector wave function (probability density function), convolved with a 'template function'

\[ \dot{P}[\psi_i] = \frac{\lambda^2}{2 \pi} \int d^3p \, |\psi_i(\vec{p}; \vec{p}_0)|^2 \, \mathcal{T}_{\mathrm{rel}}(\vec{p}) \, . \]
First-quantised model \[ \hspace{-3em} \mathcal{T}_{\mathrm{rel}}^{\mathrm{(1st)}}(\vec{p}) = \frac{1}{4} \left( 1 - \frac{M_g^4}{M_e^4} \right) \sqrt{\vec{p}^2 + M_e^2} \, , \]
Second-quantised model \[ \hspace{1em} \mathcal{T}_{\mathrm{rel}}^{\mathrm{(2nd)}}(\vec{p}) = \frac{1}{4} \left( 1 - \frac{M_g^4}{M_e^4} \right) \frac{M_e^2}{\sqrt{\vec{p}^2 + M_e^2}} \, . \]

Template functions for different localisations

$E/m = 0.001$

where $m$ is the rest mass, i.e. $M_g \equiv m$ and $M_e \equiv m + E$.

Template functions for different localisations

$E/m = 0.001$

where $m$ is the rest mass, i.e. $M_g \equiv m$ and $M_e \equiv m + E$.

Template functions for different localisations

$E/m = 10$

where $m$ is the rest mass, i.e. $M_g \equiv m$ and $M_e \equiv m + E$.

Transition rate for detector at rest

Assume detector has Gaussian profile

\[ \psi_i(\vec{p}; \vec{p}_D) = \left( \frac{L^2}{2 \pi} \right)^{3/4} \exp\left(-\frac{L^2}{4} |\vec{p} - \vec{p}_D|^2\right) \, . \]

Can obtain analytic results for detector at rest $\vec{p}_D = \vec{0}$

\[ \dot{P}_{\mathrm{rel.}}^{\mathrm{(1st)}}[\psi_i] \sim e^{\frac{L^2 M_e^2}{4}} K_1\left( \frac{L^2 M_e^2}{4} \right) \, , \quad \dot{P}_{\mathrm{rel.}}^{\mathrm{(2nd)}}[\psi_i] \sim U\left( \frac{1}{2},0,\frac{L^2 M_e^2}{2} \right) \, . \]

Transition rate for detector at rest

$L/\lambda_c = 10$

where $\lambda_c \equiv m^{-1}$ is the Compton wavelength of the detector.

Transition rate for detector at rest

$L/\lambda_c = 0.1$

where $\lambda_c \equiv m^{-1}$ is the Compton wavelength of the detector.

Transition rate for highly localised detector

  • Transition rate approximately given by template function
    •    
  • Q. What would one observe if this experiment were performed?

Open Questions

  • Q. What is a 'particle'?
  • A. A 'particle' is what a 'particle detector' detects.
    •  
  • Q. What is a 'particle detector'?
  • A. A 'particle detector' is a localised system with which one may perform local operations on a quantum field.
    •  
  • Q. What is meant by a 'localised system'? For a relativistic particle detector, how should one define its localisation?
  • A. ...

How should we interpret these results?

Localisation in RQM I

Newton and Wigner [10] defined a relativistic position operator
by requiring it satisfy a set of invariance conditions

  1. Superposition of localised systems is also localised.
  2. Invariant wrt spatial rotations and reflections.
  3. Localised states are orthogonal wrt translated states.
  4. Generators of Lorentz group applicable to the localised states.
    5.  

Coincides with the non-relativistic position operator [11]


[10] T. D. Newton and E. P. Wigner, Rev. Mod. Phys. 21, 400 (1949).
[11] R. Fong and E. G. P. Rowe, Ann. Phys. 46, 559 (1968).

How should we interpret these results?

Localisation in RQM II

Philips [12] defined a Lorentz-invariant position operator

  1. Superposition of localised systems is also localised.
  2. Invariant wrt spatial rotations and reflections.
  3. Localised states are invariant wrt Lorentz boosts.
  4. The localised states are normalisable.
  5. No subset of the set of localised states satisfies the above conditions.
    6.  

In second-quantised formalism, Philips' position state is equivalent to the action of field operator on vacuum $| \vec{x}_p \rangle \equiv \hat{\psi}(\vec{x}) | 0 \rangle$


[12] T. O. Philips, Phys. Rev. 136, B893 (1964).

How should we interpret these results?

Localisation in RQM III

Localisation schemes not first- versus second-quantised.
Correspond to different representations with associated wave equations

  • Newton-Wigner: relativistic Schrödinger equation
$$i \partial_t \psi_{\text{NW}}(x) = \sqrt{-\nabla^2 + m^2} \psi_{\text{NW}}(x) $$
  • Philips: Klein-Gordon equation (spin-0)
$$\left( \square + m^2 \right) \psi(x) = 0 $$

Two sides to relativistic localisation

Q. How should we interpret the results of these two localisations?

A. It's unclear.

  • Expect the more fundamental second-quantised model to be correct, but its "position operator" is non-Hermitian.
  • Operators in Newton-Wigner representation agree with what we expect, but no Lorentz invariance for localisation and causality issues present.
  • Is the Newton-Wigner representation the result of course-graining,
    or does it have a more fundamental status?

Summary

  • Localisation problem in UDW model / idealised atom-light interaction
    •  
  • Surprising disagreement between first- and second-quantised models!
    •  
  • Experimentally distinguishable for detectors boosted wrt lab frame

Thank you!


Evan Gale | e.gale@uq.edu.au