Entropy and area
Open accessAbstract
The ground-state density matrix for a massless free field is traced over the degrees of freedom residing inside an imaginary sphere; the resulting entropy is shown to be proportional to the area (and not the volume) of the sphere. Possible connections with the physics of black holes are discussed.
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TL;DR
Tracing the ground state of a massless free scalar field over the degrees of freedom inside an imaginary sphere yields an entropy proportional to the sphere’s area, not its volume. The result is analogous to the black-hole entropy formula \(S_{BH}=\frac14 M_{Pl}^2 A\).
Key findings
- For a massless free scalar field, the entropy from tracing over the inside of a sphere is \(S=\kappa M^2 A\), where \(A=4\pi R^2\), \(M\) is the ultraviolet cutoff, and \(\kappa\) depends only on the definition of \(M\).
- Numerical results for \(N=60\) and \(1\le n\le 30\) fit a straight line in \(R^2\); for fixed \(n\le \frac12 N\), \(S(n,N)\) is independent of \(N\) to within 0.5% for \(N=20,40,60\).
- In \(d=2\), the expected area law becomes \(S=\kappa MR\), confirmed numerically; in \(d=1\), the numerical results indicate \(S=\kappa_1\log(MR)+\kappa_2\log(\mu R)\).
- The result resembles \(S_{BH}=\frac14 M_{Pl}^2 A\), suggesting \(S\sim A\) may be more general than previously realized.
Methods
- The paper first analyzes two coupled harmonic oscillators, then generalizes to \(N\) coupled harmonic oscillators, and finally applies the result to a massless free scalar field.
- The field is regulated by a radial lattice with spacing \(a\), so the ultraviolet cutoff is \(M=a^{-1}\), and by a spherical box of radius \(L=(N+1)a\), so the infrared cutoff is \(\mu=L^{-1}\).
- Entropy is computed numerically by tracing the ground state over the first \(n\) sites and summing partial waves: \(S(n,N)=\sum_l(2l+1)S_l(n,N)\).
Limitations
- The paper states that for \(d\ge4\), radial-lattice regularization is insufficient because the sum over partial waves does not converge; a full \(d\)-dimensional lattice would increase computational complexity.
- The paper states that the linear area law cannot continue to \(n=N\), since tracing over all degrees of freedom must give \(S=0\); hence the restriction \(n\le\frac12 N\).
- The paper states that in \(d=1\), the introductory arguments break down and the numerical results show dependence on the infrared cutoff \(\mu\). Evident from the provided text: many equations appear as “formula-not-decoded,” so derivation details cannot be checked from this copy.
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Claims the paper makes
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- Findingown data§ Entropy and Area ∗ · p. 7
The computed values of S(n,N) for N = 60 and 1 ≤ n ≤ 30 are fit by a straight line as a function of R^2.
“The computed values of S ( n, N ) are shown for N = 60 and 1 ≤ n ≤ 30 as a function of R 2 in Fig. 1. As can be seen, the points are beautifully fit by a straight line:”
Quote in the paperSubstantial claim - Findingnone given§ Entropy and Area ∗ · p. 8
For d ≥ 4, regularization by a radial lattice is insufficient because the sum over partial waves does not converge.
“For d ≥ 4, regularization by a radial lattice turns out to be insufficient; the sum over partial waves does not converge.”
Quote in the paperNot a substantial claim - Findingcitation§ Entropy and Area ∗ · p. 8
For d = 1, numerical results indicate that the entropy is S = κ1 log(MR) + κ2 log(μR).
“In fact, the numerical results indicate that S = κ 1 log( MR )+ κ 2 log( µR ) in one dimension; for the first time, we see a dependence on the infrared cutoff µ [6].”
Quote in the paperSubstantial claim0 of 1 sources checked- cites [6] [6] This formula bears no apparent relation to that for the entropy of a one-dimensional black hole; see G. W. Gibbons and M. J. Perry, Int. J. Mod. Phys. D1 , 335 (1992); C. Nappi and A. Pasquinucci, Mod. Phys. Lett. A7 , 3337 (1992). - source
Not checked in the free run.
- Findingreasoning§ Entropy and Area ∗ · p. 6
The entropy associated with the ρ_out of Eq. (13) is S = ∑_i S(ξ_i), where S(ξ) is given by Eq. (6) and ξ_i = β'_i/[1 + (1 - β'^2_i)^{1/2}].
“Therefore, the entropy associated with the ρ out of Eq. (13) is just S = ∑ i S ( ξ i ), where S ( ξ ) is given by Eq. (6), and ξ i = β ′ i /[ 1 + (1 -β ′ 2 i ) 1 / 2 ] .”
Quote in the paperSubstantial claim - Findingcitation§ Entropy and Area ∗ · p. 8
For d = 2, numerical results confirm that the entropy is S = κ M R, where R is the radius of the dividing circle.
“For d = 2, our introductory arguments would lead us to expect that S = κMR , since the relevant 'area' is the circumference of the dividing circle of radius R . This is confirmed by the numerical results, which will be presented in detail elsewhere [5].”
Quote in the paperSubstantial claim0 of 1 sources checked- cites [5] [5] M. Srednicki, in preparation. -
Not checked in the free run.
- Findingreasoning§ Entropy and Area ∗ · p. 5
Eq. (5) implies that ρ_out is equivalent to a thermal density matrix for a single harmonic oscillator specified by frequency α and temperature T = α/log(1/ξ).
“Eq. (5) imples that ρ out is equivalent to a thermal density matrix for a single harmonic oscillator specified by frequency α and temperature T = α/ log(1 /ξ ).”
Quote in the paperSubstantial claim - Findingreasoning§ Entropy and Area ∗ · p. 7
Eqs. (20) and (21) demonstrate that the sum over l will converge.
“Eqs. (20) and (21) demonstrate that the sum over l will converge, and also provide a useful check on the numerical results.”
Quote in the paperNot a substantial claim - Interpretationown data§ Entropy and Area ∗ · p. 3
The amount of missing information represented by S_BH is about right, in the sense that the same answer would be obtained in the vacuum of flat space if access to the interior of a sphere with surface area A were denied and the ultraviolet cutoff set to order M_Pl.
“It shows that the amount of missing information represented by S BH is about right, in the sense that we would get the same answer in the vacuum of flat space if we did not permit ourselves access to the interior of a sphere with surface area A , and set the ultraviolet cutoff to be of order M Pl (perfectly reasonable for comparison with a quantum theory that includes gravity).”
Quote in the paperSubstantial claim - Premisecitation§ Entropy and Area ∗ · p. 3
After a black hole shrinks to negligible size, S_HR is a number of order one (depending on masses and spins of elementary particles) times the original black hole entropy.
“One finds, after the black hole has shrunk to negligible size, that S HR is a number of order one (depending on the masses and spins of the elementary particles) times the original black hole entropy [3].”
Quote in the paperSubstantial claim0 of 1 sources checked- cites [3] [3] W. H. Zurek, Phys. Rev. Lett. 49 , 1683 (1982); D. N. Page, Phys. Rev. Lett. 50 , 1013 (1983). -
Couldn’t be matched to a paper in Crossref, so it can’t be checked.
- Interpretationown data§ Entropy and Area ∗ · p. 3
Getting S ∼ A does not require the boundary of the inaccessible region to be dynamical.
“Furthermore, getting S ∼ A clearly does not require the boundary of the inaccessible region to be dynamical, since in our case it is entirely imaginary.”
Quote in the paperNot a substantial claim - Interpretationreasoning§ Entropy and Area ∗ · p. 3
The result S ∼ A is a much more general formula than has heretofore been realized.
“The new result quoted above indicates that S ∼ A is a much more general formula than has heretofore been realized.”
Quote in the paperNot a substantial claim - Interpretationreasoning§ Entropy and Area ∗ · p. 3
The fact that S_BH ≃ S_HR lends support to the idea that S_BH is related to a counting of quantum states.
“This calculation of S HR is done by counting quantum states, and the fact that S BH ≃ S HR lends support to the idea that S BH should also be related to a counting of quantum states.”
Quote in the paperSubstantial claim
- May combine two claims. 7
- Classified as probably not a substantial claim. 1
May combine two claims.
Findingown data§ Entropy and Area ∗ · p. 8For fixed n with n ≤ 1/2 N, the entropy S(n,N) is independent of N (and hence of the infrared cutoff); values are identical within 0.5% for N = 20, 40, and 60.
“Furthermore, S ( n, N ) turns out to be independent of N (and hence the value of the infrared cutoff). Specifically, for fixed n , with n ≤ 1 2 N , the values of S ( n, N ) turn out to be identical (in the worst case, to within 0.5%) for N = 20, 40, and 60.”
Quote in the paperSubstantial claimMay combine two claims.
Findingown data§ Entropy and Area ∗ · p. 1For a massless free scalar field in its ground state, tracing over the degrees of freedom inside an imaginary sphere of radius R yields an entropy S = κ M^2 A, where A = 4πR^2 and κ is a numerical constant depending on the definition of M; thus S is proportional to area, not volume.
“The ground state density matrix for a massless free field is traced over the degrees of freedom residing inside an imaginary sphere; the resulting entropy is shown to be proportional to the area (and not the volume) of the sphere.”
- Also stated in: § Entropy and Area ∗ · p. 3 · plain
“In fact, as will be shown below, S = κM 2 A , where κ is a numerical constant which depends only on the precise definition of M that we adopt.”
- Also stated in: § Entropy and Area ∗ · p. 8 · plain
“To summarize, a straightforward counting of quantum states in a simple, well-defined context has produced an entropy proportional to the surface area of the inaccessible region, inaccessible in the sense that we ignore the information contained there.”
Quote in the paperSubstantial claim- Also stated in: § Entropy and Area ∗ · p. 3 · plain
May combine two claims.
Findingcitation§ Entropy and Area ∗ · p. 3The density matrices ρ_in and ρ_out have the same eigenvalues (up to extra zeroes for the larger if ranks differ), so S' = S.
“However, it is straightforward to show that ρ in and ρ out have the same eigenvalues (with extra zeroes for the larger, if they have different rank), so that in fact S ′ = S [1].”
Quote in the paperSubstantial claim0 of 1 sources checked- cites [1] [1] Let | 0 〉 = ∑ ia ψia | i 〉 in | a 〉 out , so that ( ρ in ) ij = ( ψψ † ) ij and ( ρ out) ab = ( ψ T ψ ∗ ) ab . Now it is clear that Tr ρ k in = Tr ρ k out for any positive integer k . This can only be true if ρ in and ρ out have the same eigenvalues, up to extra zeroes. -
Couldn’t be matched to a paper in Crossref, so it can’t be checked.
May combine two claims.
Findingreasoning§ Entropy and Area ∗ · p. 8The restriction to n ≤ 1/2 N is necessary because the linear behavior cannot continue to n = N, where S = 0; S must start falling as R approaches the wall of the box at radius L = (N+1)a.
“The restriction to n ≤ 1 2 N is necessary, since the linear behavior in Fig. 1 cannot continue all the way to n = N : at this point we will have traced over all the degrees of freedom, and must find S = 0. S must therefore start falling as R begins to approach the wall of the box at radius L = ( N +1) a .”
Quote in the paperSubstantial claimMay combine two claims.
Findingreasoning§ Entropy and Area ∗ · p. 5In general β and γ do not commute, which implies that Eq. (11) is not equivalent to a thermal density matrix for a system of oscillators.
“In general β and γ will not commute, which implies that Eq. (11) is not equivalent to a thermal density matrix for a system of oscillators.”
Quote in the paperSubstantial claimMay combine two claims.
Findingreasoning§ Entropy and Area ∗ · p. 7For l ≫ N, S_l(n,N) is independent of N and is given by Eqs. (20) and (21).
“The result is that, for l ≫ N , Sl ( n, N ) is independent of N , and is given by”
Quote in the paperSubstantial claimMay combine two claims.
Findingreasoning§ Entropy and Area ∗ · p. 7H_lm is independent of m, so S_lm(n,N) = S_l(n,N); summing over m yields a factor of 2l+1, giving S(n,N) = ∑_l (2l+1) S_l(n,N).
“As can be seen from Eq. (18), Hlm is actually independent of m , and therefore so is Slm ( n, N ) = Sl ( n, N ). Summing over m just yields a factor of 2 l + 1, and so we have S ( n, N ) = ∑ l (2 l + 1) Sl ( n, N ).”
Quote in the paperSubstantial claimClassified as probably not a substantial claim.
Premisecitation§ Entropy and Area ∗ · p. 3The intrinsic entropy of a black hole is S_BH = 1/4 M_Pl^2 A, where M_Pl is the Planck mass and A is the horizon surface area.
“This result bears a striking similarity to the formula for the intrinsic entropy of a black hole, S BH = 1 4 M 2 Pl A , where M Pl is the Planck mass and A is the surface area of the horizon of the black hole [2].”
Quote in the paperSubstantial claim0 of 1 sources checked- cites [2] [2] J. D. Bekenstein, Phys. Rev. D 7 , 2333 (1973); Phys. Rev. D 9 , 3292 (1974); S. W. Hawking, Comm. Math. Phys. 43 , 199 (1975); G. W. Gibbons and S. W. Hawking, Phys. Rev. D 15 , 2752 (1977); S. W. Hawking, in General Relativity: An Einstein Centenary Survey , ed. S. W. Hawking and W. Israel (Cambridge, 1979); R. Kallosh, T. Ort´ ın, and A. Peet, Stanford Univ. Report SU-ITP-92-29 (hep-th/9211015). -
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For critical regimes, the block entanglement entropy S_L in spin chains coincides with the entropy of a subregion in conformal field theory.
“Our results establish a precise connection between concepts of quantum information, condensed matter physics, and quantum field theory, by showing that the behavior of critical entanglement in spin systems is analogous to that of entropy in conformal field theories.”
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- Authors
- Srednicki, M.
- Affiliations
- Lawrence Berkeley National Laboratory (US)
- Corresponding author
- Srednicki, Mark
- ORCID
- Srednicki, Mark
- Venue
- Physical Review Letters
- Publisher
- American Physical Society (APS)
- Published
- 1993-08-02
- Volume
- 71
- Issue
- 5
- Pages
- 666-669
- ISSN
- 0031-9007 (ISSN-L)
- OpenAlex
- W2053387157
- Semantic Scholar
- b065054640fd7a7daaa4d880afcc47cad8eea8a8
- Cited by
- 1,858
- References
- 11
- Citations by year
- 2026: 61 · 2025: 90 · 2024: 90 · 2023: 88 · 2022: 101 · 2021: 124 · 2020: 94 · 2019: 117 · 2018: 77 · 2017: 94 · 2016: 107 · 2015: 100 · 2014: 111 · 2013: 87 · 2012: 59
- FWCI
- 18.95
- Type
- journal article
- Language
- en
- Open access
- green
- arxiv.org
- Locations
- Physical Review Letters (publishedVersion)
- PubMed (publishedVersion)
- arXiv (Cornell University) (submittedVersion)
- arXiv (Cornell University)
- Topics
- Physical Sciences › Physics and Astronomy › Nuclear and High Energy Physics › Black Holes and Theoretical Physics
- Physical Sciences › Physics and Astronomy › Statistical and Nonlinear Physics › Statistical Mechanics and Entropy
- Physical Sciences › Physics and Astronomy › Atomic and Molecular Physics, and Optics › Quantum many-body systems
- Keywords
- entropy-area relation, area law, black holes, quantum field theory in curved spacetime, holography
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