Idée - Computational Physics - # Maxwell Field with Gauge Fixing Term in Expanding Universe

Exact Solutions and Stress Tensor of the Maxwell Field with Gauge Fixing Term in the Radiation-Dominant and Matter-Dominant Stages of the Expanding Universe

Q: How would the results change if the Maxwell field was coupled to other fields, such as a scalar field or a fermion field?

If the Maxwell field were coupled to other fields, such as a scalar field or a fermion field, the results would likely differ significantly due to the interactions introduced by these couplings. For instance, when coupling to a scalar field, the dynamics of the Maxwell field would be modified by the scalar field's potential and kinetic terms, potentially leading to nontrivial interactions that could affect the stress tensor components. The presence of a scalar field could introduce additional UV divergences in the stress tensor, necessitating different regularization techniques compared to the pure Maxwell field case. In the case of fermionic fields, the coupling would introduce spinor degrees of freedom, which could lead to a richer structure in the stress tensor. The fermionic contributions would also need to be treated with care, particularly in curved spacetimes, where the spinor fields must be defined appropriately to maintain covariance. The resulting stress tensor would likely exhibit different behaviors under gauge transformations and could contribute to phenomena such as fermionic vacuum polarization, which might alter the effective stress-energy content of the universe.

Q: What are the implications of the vanishing GF stress tensor for models of dark energy and the cosmological constant problem?

The vanishing gauge fixing (GF) stress tensor has significant implications for models of dark energy and the cosmological constant problem. Since the GF stress tensor is shown to be zero in both the radiation-dominant (RD) and matter-dominant (MD) stages, it cannot contribute to the energy density of the universe. This finding challenges the notion that the GF term could serve as a candidate for dark energy, which is often invoked to explain the accelerated expansion of the universe. Moreover, the results indicate that the regularized vacuum stress tensor of the Maxwell field does not yield a non-zero contribution to the cosmological constant. This reinforces the idea that the cosmological constant problem, which arises from the discrepancy between the observed value of dark energy and theoretical predictions from quantum field theory, cannot be resolved by invoking the GF stress tensor. Instead, the results suggest that alternative mechanisms or fields must be considered to account for dark energy, as the GF term does not provide a viable solution.

Q: Can the techniques used in this analysis be extended to study the behavior of other gauge fields, such as non-Abelian gauge fields, in curved spacetimes?

Yes, the techniques employed in this analysis can be extended to study the behavior of other gauge fields, including non-Abelian gauge fields, in curved spacetimes. The framework of covariant canonical quantization, as applied to the Maxwell field with a gauge fixing term, can similarly be adapted to non-Abelian gauge theories. The key steps, such as deriving the equations of motion, implementing gauge fixing, and calculating the stress tensor, would follow analogous procedures. However, non-Abelian gauge fields introduce additional complexities, such as self-interactions and the need to account for the structure of the gauge group. The presence of non-Abelian gauge fields may lead to more intricate vacuum structures and potential UV divergences that differ from those encountered in Abelian theories. The regularization techniques would need to be carefully tailored to address these complexities, potentially requiring higher-order regularization schemes or different approaches to handle the non-linearities inherent in non-Abelian gauge theories. Overall, while the foundational techniques are applicable, the specific details and outcomes would depend on the nature of the non-Abelian gauge fields and their interactions with the background spacetime.

Concepts de base

The Maxwell field with a general gauge fixing term is studied in the radiation-dominant and matter-dominant stages of the expanding universe. Exact solutions are derived, covariant canonical quantization is performed, and the stress tensor is obtained in the Gupta-Bleuler physical states. The regularized vacuum stress tensor is found to be zero, independent of the gauge fixing constant, and there is no trace anomaly.

Résumé

The authors study the Maxwell field with a general gauge fixing (GF) term in the radiation-dominant (RD) and matter-dominant (MD) stages of the expanding universe. They derive the exact solutions for the longitudinal and temporal components of the Maxwell field, which are mixed up due to the GF term. The covariant canonical quantization is then implemented, and the stress tensor is calculated in the Gupta-Bleuler (GB) physical states.

The key findings are:

The transverse stress tensor has both particle and vacuum parts, with the vacuum part containing only a UV divergent k^4 term that becomes zero after 0th-order adiabatic regularization.
The longitudinal-temporal (LT) stress tensor is zero in the GB states due to a cancelation between the longitudinal and temporal parts, for both the particle and vacuum contributions.
The particle part of the GF stress tensor is also zero in the GB states.
The vacuum GF stress tensor exhibits different behaviors in the RD and MD stages - in the RD stage it contains k^4 and k^2 divergences and becomes zero by 2nd-order regularization, while in the MD stage it contains k^4, k^2, and k^0 divergences and requires 4th-order regularization.
In both the RD and MD stages, the total regularized vacuum stress tensor is zero, independent of the GF constant, and there is no trace anomaly. Only the transverse photon part remains.
The GF stress tensor, including both particle and vacuum parts, is zero, so there is no need to introduce a ghost field to cancel it, as has been suggested in some previous works.

The authors conclude that the vanishing GF stress tensor cannot be a candidate for dark energy, and that the trace anomaly arises as an artifact from improper use of higher-order regularization on massive fields, rather than occurring for massless fields directly.

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Stats

The scale factor in the radiation-dominant stage is a(τ) = a_r τ, where a_r is a constant.
The scale factor in the matter-dominant stage is a(τ) = a_m τ^2, where a_m is a constant.

Citations

"The transverse stress tensor has both particle and vacuum parts, with the vacuum part containing only a UV divergent k^4 term that becomes zero after 0th-order adiabatic regularization."
"The longitudinal-temporal (LT) stress tensor is zero in the GB states due to a cancelation between the longitudinal and temporal parts, for both the particle and vacuum contributions."
"In both the RD and MD stages, the total regularized vacuum stress tensor is zero, independent of the GF constant, and there is no trace anomaly."

Idées clés tirées de

Maxwell field with gauge fixing term in the radiation- and matter-dominant stages: exact solution and stress tensor

by Xuan Ye, Yan... à arxiv.org 10-02-2024

https://arxiv.org/pdf/2301.06124.pdf

Maxwell field with gauge fixing term in the radiation- and matter-dominant stages: exact solution and stress tensor

Questions plus approfondies

How would the results change if the Maxwell field was coupled to other fields, such as a scalar field or a fermion field?

If the Maxwell field were coupled to other fields, such as a scalar field or a fermion field, the results would likely differ significantly due to the interactions introduced by these couplings. For instance, when coupling to a scalar field, the dynamics of the Maxwell field would be modified by the scalar field's potential and kinetic terms, potentially leading to nontrivial interactions that could affect the stress tensor components. The presence of a scalar field could introduce additional UV divergences in the stress tensor, necessitating different regularization techniques compared to the pure Maxwell field case.
In the case of fermionic fields, the coupling would introduce spinor degrees of freedom, which could lead to a richer structure in the stress tensor. The fermionic contributions would also need to be treated with care, particularly in curved spacetimes, where the spinor fields must be defined appropriately to maintain covariance. The resulting stress tensor would likely exhibit different behaviors under gauge transformations and could contribute to phenomena such as fermionic vacuum polarization, which might alter the effective stress-energy content of the universe.

What are the implications of the vanishing GF stress tensor for models of dark energy and the cosmological constant problem?

The vanishing gauge fixing (GF) stress tensor has significant implications for models of dark energy and the cosmological constant problem. Since the GF stress tensor is shown to be zero in both the radiation-dominant (RD) and matter-dominant (MD) stages, it cannot contribute to the energy density of the universe. This finding challenges the notion that the GF term could serve as a candidate for dark energy, which is often invoked to explain the accelerated expansion of the universe.
Moreover, the results indicate that the regularized vacuum stress tensor of the Maxwell field does not yield a non-zero contribution to the cosmological constant. This reinforces the idea that the cosmological constant problem, which arises from the discrepancy between the observed value of dark energy and theoretical predictions from quantum field theory, cannot be resolved by invoking the GF stress tensor. Instead, the results suggest that alternative mechanisms or fields must be considered to account for dark energy, as the GF term does not provide a viable solution.

Can the techniques used in this analysis be extended to study the behavior of other gauge fields, such as non-Abelian gauge fields, in curved spacetimes?

Yes, the techniques employed in this analysis can be extended to study the behavior of other gauge fields, including non-Abelian gauge fields, in curved spacetimes. The framework of covariant canonical quantization, as applied to the Maxwell field with a gauge fixing term, can similarly be adapted to non-Abelian gauge theories. The key steps, such as deriving the equations of motion, implementing gauge fixing, and calculating the stress tensor, would follow analogous procedures.
However, non-Abelian gauge fields introduce additional complexities, such as self-interactions and the need to account for the structure of the gauge group. The presence of non-Abelian gauge fields may lead to more intricate vacuum structures and potential UV divergences that differ from those encountered in Abelian theories. The regularization techniques would need to be carefully tailored to address these complexities, potentially requiring higher-order regularization schemes or different approaches to handle the non-linearities inherent in non-Abelian gauge theories.
Overall, while the foundational techniques are applicable, the specific details and outcomes would depend on the nature of the non-Abelian gauge fields and their interactions with the background spacetime.