https://academic.oup.com/ptep/article/2019/4/043B06/5482208
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பிரபஞ்சத்தின் சன்னலுக்கு வெளியே
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இந்த பிரபஞ்சத்தின் கடைக்கோடி சன்னல் தான் கருந்துளை எனும் ப்ளாக் ஹோல்.இந்த சன்னலை எட்டிப்பார்ப்பதும் இயலாது.நம் மின்காந்தக்கதிர் இழையும் ஒளிக்கதிர் கொண்டு தான் அங்கு துழாவ முடியும்.
அப்படி ஊடுருவும்போது அந்த ஒளிக்கதிர் கருந்துளையால் விழுங்கப்பட்டு விடும்.அப்படி அந்த சன்னலைத்தாண்டிய பிரபஞ்சத்தை (இது நமதிலிருந்து வேறு பட்டது)யும் அதன் வடிவகணிதத்தையும் கணிக்கும் ஒரு நுட்ப சமன்பாடு மூலம் துருவிப்பார்க்கும் அந்த ஆச்சரியம் நிறைந்த கண்கள் நம் விஞ்ஞானிகளின் மூளைச்செதில்களின் நியூரான்களுக்கு உண்டு.அதைப்பற்றிய கட்டுரையே இது.
நம் பிரபஞ்சம் இப்போது அதிர்விழையக்கோட்பாட்டால் (ஸ்ட்ரிங் தியரி) அணுகப்படுகிறது.நம் அடிப்படை ஆற்றல்களில் ஈர்ப்பை தவிர மற்ற மூன்றின் அதாவது மின்காந்தம்,வலுவற்ற ஆற்றல்,மற்றும் வலுமிகு அணு ஆற்றல் இவைகளின் "ஒன்றிய நிலைப்பாடு"(யுனிஃபிகேஷன்) எனும் இந்த நுண்மாண்னுழைபுலம் புகுந்து நம்மை நம்மை நம்மீது நாமே ஒளிபாய்ச்சிக்கொள்வோம்.இந்த ஒளிவீச்சு எனும் ப்ரொஜெக்டிவிஸம் மட்டுமே நம்மை காக்க இயலும்.
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Comments on the Stress-Energy Tensor Operator in Curved Spacetime
Communications in Mathematical Physics 232, 189–221 (2003)
Abstract:
The technique based on a *-algebra of Wick products of field operators in curved spacetime, in the local covariant version proposed by Hollands and Wald, is strightforwardly generalized in order to define the stress-energy tensor operator in curved globally hyperbolic spacetimes. In particular, the locality and covariance requirement is generalized to Wick products of differentiated quantum fields. Within the proposed formalism, there is room to accomplish all of the physical requirements provided that known problems concerning the conservation of the stress-energy tensor are assumed to be related to the interface between the quantum and classical formalism. The proposed stress-energy tensor operator turns out to be conserved and reduces to the classical form if field operators are replaced by classical fields satisfying the equation of motion. The definition is based on the existence of convenient counterterms given by certain local Wick products of differentiated fields. These terms are independent from the arbitrary length scale (and any quantum state) and they classically vanish on solutions of the Klein-Gordon equation. Considering the averaged stress-energy tensor with respect to Hadamard quantum states, the presented definition turns out to be equivalent to an improved point-splitting renormalization procedure which makes use of the nonambiguous part of the Hadamard parametrix only that is determined by the local geometry and the parameters which appear in the Klein-Gordon operator. In particular, no extra added-by-hand term g αβQ and no arbitrary smooth part of the Hadamard parametrix (generated by some arbitrary smooth term ``ω 0 '') are involved. The averaged stress-energy tensor obtained by the point-splitting procedure also coincides with that found by employing the local ζ-function approach whenever that technique can be implemented.
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Comments on the Stress-Energy Tensor Operator in Curved Spacetime | SpringerLink
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In this paper, we have constructed the holographic space from the primary scalar field in a free massless O(n) vector model by a flow equation at the next-to-leading order in the 1/n expansion. We investigated some properties of the bulk theory by calculating the induced Einstein tensor in the 1/n expansion. After defining pre-geometric operators, we have calculated the VEV of the Einstein tensor operator at the NLO in the 1/n expansion. As a result, the NLO correction appeared in the VEV of the Einstein operator but not in the induced metric gAB. We therefore regarded the NLO correction of the induced Einstein operator as the quantum correction to the cosmological constant from the dual gravity theory on AdS space, which is proposed as the free higher-spin theory [38]. We have also shown that the stress–energy tensor for the vacuum state is covariant and proportional to gAB, thanks to the conformal symmetry of the boundary theory, which turns into the bulk isometry of the AdS space.
In our approach, the bulk AdS radial direction emerges as the smearing scale for the boundary CFT. It is important to clarify how to determine the whole structure of the bulk theory. The bulk stress–energy tensor corresponding to the vacuum state calculated in this paper may give a hint to constructing the bulk theory.
In this paper we computed the one-loop correction to the cosmological constant in the bulk theory, which is supposed to be the free higher-spin theory [38], from the dual free O(n) vector model in the proposed framework. On the other hand, one-loop tests of higher-spin/vector model duality have already appeared in Refs. [39,40] (see also Refs. [41–45]). Their intriguing result is that the logarithmic divergence in the one-loop correction to the free energy cancels for higher-spin gauge theories employing a standard zeta function regularization in accordance with the analysis of sphere partition functions of their dual vector models. In particular, it was confirmed that the finite part vanishes for a free higher-spin theory, with its scalar field obeying the standard boundary condition. In the flow equation approach, the one-loop correction to the cosmological constant, which presumably corresponds to that of the vacuum energy, is free from UV divergence without specifying any regularization scheme. This is because the computation of the one-loop correction reduces to the two-point function of the flowed field, which has no UV divergence by construction. Our result for the finite part correction to the cosmological constant for a free higher-spin theory does not vanish in any dimension. It is highly important to fill the gap between these results, which could be done by identifying a bulk local operator in the flow equation approach.
The next important step is to evaluate the bulk stress–energy tensor corresponding to excited states. Indeed, we can easily generalize the computation of the VEV for the Einstein operator presented in this paper to that of arbitrary states as follows. We consider a set of states {|O⟩} in CFT with the inner product ⟨O|O′⟩=δO,O′, where the meaning of δO,O′ depends on the type of states. Then we evaluate the matrix element of the Einstein operator in the 1/n expansion by using the covariant perturbation given in Appendix A as
⟨O|G^AB|O′⟩==⟨O|GAB|O′⟩+⟨O|G˙AB|O′⟩+⟨O|G¨AB|O′⟩+⋯{GAB+⟨0|G¨AB|0⟩}δO,O′+⟨O|G˙AB|O′⟩+⟨O|G¨AB|O′⟩c+O(1n2),
(5.1)
where ⟨O|X^|O′⟩c:=⟨O|X^|O′⟩−⟨0|X^|0⟩δO,O′ for an arbitrary operator X^. As asserted in Sect. 3, we interpret the matrix element of the Einstein operator as the bulk stress–energy tensor through Eq. (3.1), which we may call the quantum Einstein equation. It is natural to interpret in this way that the corresponding bulk stress–energy tensor consists of the cosmological constant and the contribution from the matter field in the bulk:
TbulkAB=−ΛgmatAB+TmatAB,gmatAB:=⟨O|g^AB|O′⟩.
(5.2)
Notice that we have already calculated the first term in Eq. (5.1) as
GAB+⟨0|G¨AB|0⟩=−ΛgAB,Λ=−d(d−1)2L2Δ(1+d+4n)+⋯,
(5.3)
which represents the vacuum contribution. Therefore, the contribution of the matter field to the bulk stress–energy tensor is given by
TmatAB=⟨O|G˙AB|O′⟩+⟨O|G¨AB|O′⟩c+Λ⟨O|g^AB|O′⟩c.
(5.4)
It is very important to compute this bulk stress–energy tensor in the construction of the dual bulk theory beyond the vacuum or geometry level. We are currently calculating TmatAB , and will report the result elsewhere.
This program can be extended to the case of the λφ4 theory in three dimensions. In the previous investigation [46], while the induced metric describes the AdS4 space at the leading order, the next-to-leading-order corrections make the space asymptotically AdS only in the UV and IR limits with different radii. These two limits correspond to the asymptotically free UV fixed point and the Wilson–Fischer IR fixed point of the boundary theory, respectively. It would be interesting to investigate how the stress–energy tensor in the bulk behaves from UV to IR at the NLO. We expect that this behaves in a similar manner to the one computed from the corresponding solution for the dual bulk (higher-spin) theory.
We hope to report on the progress with these issues in the near future.
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