insight - Scientific Computing - # Quantum Field Theory

Evanescent Operators in Gauge Theories: Implications for Unitarity in Non-Integer Dimensions

Q: How do these findings concerning unitarity violations in non-integer dimensions affect the interpretation and validity of calculations using dimensional regularization in quantum field theory?

This is a subtle question that touches on the heart of how we use dimensional regularization. Here's a breakdown of the implications: Dimensional regularization as a tool: It's important to remember that dimensional regularization, in many ways, is a mathematical trick. We analytically continue the spacetime dimension to regulate divergences, and then we take the limit back to integer dimensions for physical results. The findings of unitarity violation in non-integer dimensions don't directly invalidate calculations done this way. The importance of the ϵ→0 limit: The key point is that physical observables are extracted in the limit where the regulator (in this case, ϵ) goes to zero. While the theory might exhibit peculiar features like negative-norm states or complex anomalous dimensions for non-zero ϵ, these should disappear as we approach four dimensions. Potential subtleties: There are a few caveats to keep in mind: Non-perturbative effects: The paper focuses on perturbative calculations. It's possible that non-perturbative effects could change the behavior as ϵ→0. Scheme dependence: Dimensional regularization comes with choices (schemes) on how to treat certain objects (like γ5 in chiral theories). The manifestation of unitarity violation might be scheme-dependent. A reason for caution: While not invalidating dimensional regularization, these findings highlight that the procedure is not entirely benign. It's a reminder to be cautious and to carefully examine the behavior of results as we remove the regulator.

Q: Could there be alternative formulations of gauge theories in non-integer dimensions that preserve unitarity despite the presence of evanescent operators?

This is an open question and an active area of research. Here are some avenues being explored: Modifying the operator space: One possibility is that the standard way of defining the space of operators in non-integer dimensions needs to be revisited. Perhaps there's a more "natural" choice of basis that avoids the appearance of negative-norm states. This might involve: New inner products: Exploring alternative definitions of the inner product on the space of operators. Constraints from unitarity: Imposing unitarity as a guiding principle to constrain the allowed operator content. Alternative regularization schemes: Dimensional regularization is just one way to handle divergences. Exploring other schemes might lead to formulations where unitarity is more manifest. Some possibilities include: Lattice regularization: Formulating gauge theories on a discrete spacetime lattice. This approach has its own challenges in non-integer dimensions but could offer valuable insights. Causal perturbation theory: This approach emphasizes causality and avoids the explicit introduction of a regulator, potentially leading to a more unitarity-preserving framework. Rethinking spacetime: At a more fundamental level, these findings might suggest that our understanding of spacetime needs to be revised in the context of quantum gravity. Non-commutative geometry or other exotic spacetime structures could potentially resolve the tension between unitarity and non-integer dimensions.

Q: What are the broader implications of these findings for our understanding of the relationship between quantum field theory, unitarity, and the nature of spacetime?

These findings, while technical, hint at deeper questions about the foundations of quantum field theory and its interplay with spacetime: The rigidity of unitarity: Unitarity is a cornerstone of quantum mechanics, ensuring the consistency of probabilities. The fact that it seems to be challenged in non-integer dimensions raises fundamental questions: Is unitarity truly fundamental? Or is it an emergent property that only holds in certain limiting cases, like integer dimensions? What are the limits of QFT? Does this point to a breakdown of our standard quantum field theoretic description in certain regimes? Spacetime and quantum theory: The appearance of unitarity violation in non-integer dimensions suggests a deep connection between the nature of spacetime and quantum theory: Quantizing spacetime: Perhaps a quantum theory of gravity, where spacetime itself is quantized, would naturally resolve these issues. Emergent spacetime: Alternatively, these findings might support the idea that spacetime is not fundamental but emerges from an underlying quantum system. New avenues for research: These results provide a strong motivation to explore: Non-perturbative formulations: Going beyond perturbation theory to understand if unitarity is restored non-perturbatively. Connections to other approaches: Investigating how these findings connect to other approaches to quantum gravity, such as string theory or loop quantum gravity. In essence, the violation of unitarity in non-integer dimensions, while potentially an artifact of our current techniques, might be a valuable clue pointing towards a deeper understanding of quantum field theory and the quantum nature of spacetime.

Core Concepts

Gauge theories in non-integer spacetime dimensions, commonly used in dimensional regularization and the study of critical phenomena, exhibit violations of unitarity due to the presence of evanescent operators, which manifest as negative-norm states and complex anomalous dimensions.

Abstract

Bibliographic Information:

Jin, Q., Ren, K., Yang, G., & Yu, R. (2024). Gluonic evanescent operators: negative-norm states and complex anomalous dimensions. Journal of High Energy Physics. arXiv:2312.08445v2 [hep-th]

Research Objective:

This research paper investigates the role of evanescent operators in gauge theories, specifically focusing on their contribution to unitarity violations in non-integer spacetime dimensions.

Methodology:

The authors develop an efficient method for calculating the norms of gauge-invariant operators in Yang-Mills theory using on-shell form factors. They analyze the Gram matrices of these operators in general spacetime dimensions to identify negative-norm states. Additionally, they calculate the one-loop anomalous dimensions of these operators, revealing the presence of complex anomalous dimensions.

Key Findings:

The analysis reveals the existence of negative-norm states among evanescent operators in Yang-Mills theory, indicating a violation of unitarity in non-integer spacetime dimensions.
Complex anomalous dimensions, another sign of unitarity violation, are found to emerge for operators with canonical dimension 12 and higher.
The number of complex anomalous dimensions matches the number of negative-norm states for all length-4 operators examined, up to canonical dimension 16.
Similar patterns of non-unitarity, including negative-norm states and complex anomalous dimensions, are observed in Yang-Mills theory coupled with scalar fields.

Main Conclusions:

The presence of negative-norm states and complex anomalous dimensions associated with evanescent operators provides compelling evidence that general gauge theories are non-unitary in non-integer spacetime dimensions. This finding has significant implications for the understanding of quantum field theory in general dimensions and its applications in areas like dimensional regularization and the study of critical phenomena.

Significance:

This research sheds light on the subtle but crucial role of evanescent operators in gauge theories and their impact on the fundamental principle of unitarity. The findings challenge the conventional understanding of quantum field theory in non-integer dimensions and highlight the need for further investigation into the implications of these unitarity violations.

Limitations and Future Research:

The study focuses on specific types of operators and a limited range of dimensions. Further research could explore the behavior of evanescent operators in other gauge theories, higher-loop corrections, and different spacetime dimensions to gain a more comprehensive understanding of unitarity violations in quantum field theory.

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Stats

Complex anomalous dimensions appear at a relatively low mass dimension ∆0 = 12 in YM theory, contrasting with the scalar φ4 theory, where they do not appear until ∆0 = 23.
The Gram matrix of length-4 dim-12 operators is not positive definite, signaling the violation of unitarity of the theory at 4 < d < 5.
There are 107 dim-12 length-4 single-trace operators, among which 25 are evanescent.
There are 151 dim-12 length-5 single-trace operators, among which 61 are evanescent.

Quotes

Key Insights Distilled From

Gluonic evanescent operators: negative-norm states and complex anomalous dimensions

by Qingjun Jin,... at arxiv.org 11-12-2024

https://arxiv.org/pdf/2312.08445.pdf

Gluonic evanescent operators: negative-norm states and complex anomalous dimensions

Deeper Inquiries

How do these findings concerning unitarity violations in non-integer dimensions affect the interpretation and validity of calculations using dimensional regularization in quantum field theory?

This is a subtle question that touches on the heart of how we use dimensional regularization. Here's a breakdown of the implications:

Dimensional regularization as a tool: It's important to remember that dimensional regularization, in many ways, is a mathematical trick. We analytically continue the spacetime dimension to regulate divergences, and then we take the limit back to integer dimensions for physical results. The findings of unitarity violation in non-integer dimensions don't directly invalidate calculations done this way.
The importance of the ϵ→0 limit: The key point is that physical observables are extracted in the limit where the regulator (in this case, ϵ) goes to zero.  While the theory might exhibit peculiar features like negative-norm states or complex anomalous dimensions for non-zero ϵ, these should disappear as we approach four dimensions.
Potential subtleties:  There are a few caveats to keep in mind:

Non-perturbative effects: The paper focuses on perturbative calculations. It's possible that non-perturbative effects could change the behavior as ϵ→0.
Scheme dependence: Dimensional regularization comes with choices (schemes) on how to treat certain objects (like γ5 in chiral theories). The manifestation of unitarity violation might be scheme-dependent.


A reason for caution: While not invalidating dimensional regularization, these findings highlight that the procedure is not entirely benign. It's a reminder to be cautious and to carefully examine the behavior of results as we remove the regulator.

Could there be alternative formulations of gauge theories in non-integer dimensions that preserve unitarity despite the presence of evanescent operators?

This is an open question and an active area of research. Here are some avenues being explored:

Modifying the operator space: One possibility is that the standard way of defining the space of operators in non-integer dimensions needs to be revisited. Perhaps there's a more "natural" choice of basis that avoids the appearance of negative-norm states. This might involve:

New inner products: Exploring alternative definitions of the inner product on the space of operators.
Constraints from unitarity: Imposing unitarity as a guiding principle to constrain the allowed operator content.

Alternative regularization schemes: Dimensional regularization is just one way to handle divergences. Exploring other schemes might lead to formulations where unitarity is more manifest. Some possibilities include:

Lattice regularization:  Formulating gauge theories on a discrete spacetime lattice. This approach has its own challenges in non-integer dimensions but could offer valuable insights.
Causal perturbation theory: This approach emphasizes causality and avoids the explicit introduction of a regulator, potentially leading to a more unitarity-preserving framework.

Rethinking spacetime:  At a more fundamental level, these findings might suggest that our understanding of spacetime needs to be revised in the context of quantum gravity. Non-commutative geometry or other exotic spacetime structures could potentially resolve the tension between unitarity and non-integer dimensions.

What are the broader implications of these findings for our understanding of the relationship between quantum field theory, unitarity, and the nature of spacetime?

These findings, while technical, hint at deeper questions about the foundations of quantum field theory and its interplay with spacetime:

The rigidity of unitarity: Unitarity is a cornerstone of quantum mechanics, ensuring the consistency of probabilities. The fact that it seems to be challenged in non-integer dimensions raises fundamental questions:

Is unitarity truly fundamental? Or is it an emergent property that only holds in certain limiting cases, like integer dimensions?
What are the limits of QFT? Does this point to a breakdown of our standard quantum field theoretic description in certain regimes?

Spacetime and quantum theory: The appearance of unitarity violation in non-integer dimensions suggests a deep connection between the nature of spacetime and quantum theory:

Quantizing spacetime:  Perhaps a quantum theory of gravity, where spacetime itself is quantized, would naturally resolve these issues.
Emergent spacetime:  Alternatively, these findings might support the idea that spacetime is not fundamental but emerges from an underlying quantum system.

New avenues for research:  These results provide a strong motivation to explore:

Non-perturbative formulations:  Going beyond perturbation theory to understand if unitarity is restored non-perturbatively.
Connections to other approaches:  Investigating how these findings connect to other approaches to quantum gravity, such as string theory or loop quantum gravity.
In essence, the violation of unitarity in non-integer dimensions, while potentially an artifact of our current techniques, might be a valuable clue pointing towards a deeper understanding of quantum field theory and the quantum nature of spacetime.