The vacuum expectation value (VEV) of energy in different theories depends significantly on the framework and context, as well as the assumptions used in the theory’s formulation. Below, I outline the approximate vacuum energy contributions for different frameworks in physics, showing how they vary based on assumptions and scale:
- In QFT, the vacuum energy arises from summing up the zero-point energies of all quantum fields, particularly for electromagnetic fields in the case of QED.
- Estimation: Each mode of a quantized field contributes an energy of (\frac{1}{2} \hbar \omega), where (\omega) is the frequency of the mode. Summing over all modes leads to a theoretically large vacuum energy density, often yielding results on the order of: [ \rho_{\text{vac}} \sim 10^{113} , \text{J/m}^3 ]
- This theoretical value is extremely high and requires renormalization techniques to deal with divergences, where only differences in energy are physically meaningful.
- The cosmological constant (( \Lambda )) in General Relativity is often interpreted as a form of vacuum energy density that fills all of space. Observations suggest that this energy density drives the accelerated expansion of the universe.
- Estimation: Based on astronomical observations, the vacuum energy density associated with the cosmological constant is around: [ \rho_{\text{vac, cosmological}} \sim 10^{-9} , \text{J/m}^3 ]
- This value is about 120 orders of magnitude smaller than the vacuum energy density predicted by QFT, creating what’s known as the cosmological constant problem.
- In the context of the Casimir Effect, the vacuum energy is modified by boundary conditions, such as two conducting plates placed close together.
- Estimation: The Casimir energy per unit area ( A ) between two plates a distance ( d ) apart is given by: [ E_{\text{Casimir}} \approx -\frac{\pi^2 \hbar c}{240 d^3} A ]
- For plates separated by a micron ((d \sim 1 , \mu m)), this energy density is small but measurable and is on the order of (10^{-3} , \text{J/m}^3), which is much larger than the cosmological constant but still much smaller than QFT predictions.
- In supersymmetric theories, bosonic and fermionic contributions to the vacuum energy cancel each other out in a perfectly supersymmetric universe, resulting in a zero vacuum energy.
- Estimation: However, since SUSY is broken at energies above the electroweak scale (around (10^{3} , \text{GeV})), this results in a non-zero vacuum energy, though much lower than the QFT prediction without SUSY.
| Theory | Vacuum Energy Density ((\rho_{\text{vac}})) | Context |
|---|---|---|
| Quantum Field Theory | ( \sim 10^{113} , \text{J/m}^3 ) | Summing zero-point energies of fields |
| Cosmological Constant | ( \sim 10^{-9} , \text{J/m}^3 ) | Based on observed dark energy |
| Casimir Effect | ( \sim 10^{-3} , \text{J/m}^3 ) (for (d = 1 \mu m)) | Modified by boundary conditions |
| Supersymmetry (SUSY) | Potentially ( \sim 0 ) in exact SUSY, non-zero if broken | Depends on SUSY breaking scale |
The vast difference in values (especially the 120 orders of magnitude discrepancy between QFT and the cosmological constant) remains one of the great unsolved problems in theoretical physics.
Here are some key sources and references for understanding the vacuum expectation value (VEV) of energy across different theories:
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Quantum Field Theory (QFT) and Quantum Electrodynamics (QED) Vacuum Energy:
- Source: "Quantum Field Theory" by Mark Srednicki (2007), especially Chapters on the quantization of fields.
- Source: Steven Weinberg's "The Quantum Theory of Fields" (Vol. 1) – This textbook discusses the quantization of fields, zero-point energy contributions, and regularization.
- Summary: These sources explain the calculation of zero-point energy in QFT and the concept of vacuum fluctuations, leading to the high theoretical predictions of vacuum energy.
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Cosmological Constant and Observational Cosmology:
- Source: Peebles, P.J.E., and Bharat Ratra. "The cosmological constant and dark energy." Reviews of Modern Physics 75.2 (2003): 559–606. DOI: 10.1103/RevModPhys.75.559
- Source: "Introduction to Cosmology" by Barbara Ryden (2016), particularly the sections on dark energy and the cosmological constant.
- Summary: These works provide insight into the cosmological constant problem and the observed vacuum energy density associated with dark energy in cosmology.
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Casimir Effect:
- Source: Casimir, H.B.G. "On the attraction between two perfectly conducting plates." Proceedings of the Koninklijke Nederlandse Akademie van Wetenschappen 51 (1948): 793–795. Available in many physics archives.
- Source: "The Casimir Effect: Physical Manifestations of Zero-Point Energy" by K.A. Milton (2001) – This book is a comprehensive resource on the Casimir effect and vacuum fluctuations.
- Summary: Casimir's original paper and Milton's book discuss the calculation of vacuum energy changes between conducting plates, leading to the measurable Casimir force.
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Supersymmetry (SUSY) and Vacuum Energy Cancellation:
- Source: "Supersymmetry and Supergravity" by Julius Wess and Jonathan Bagger (1992) – This book covers the basics of SUSY and how boson-fermion symmetry can theoretically cancel vacuum energy.
- Source: "Supersymmetry" by P. Binétruy (2006) – This is another excellent source for understanding SUSY and its implications on vacuum energy, particularly in the context of SUSY breaking.
- Summary: Wess and Bagger’s book explains how SUSY could cancel out vacuum energy in an unbroken state and how SUSY breaking at higher energies results in a residual, lower vacuum energy.
These sources provide foundational knowledge and detailed derivations for understanding vacuum energy in different theoretical contexts and how it relates to observed and predicted values in modern physics.