6478419
“Theoretical mathematics”: toward a cultural synthesis of mathematics and theoretical physics
Is speculative mathematics dangerous? Recent interactions between physics and mathematics pose the question with some force: traditional mathematical norms discourage speculation, but it is the fabric of theoretical physics. In practice there can be benefits, but there can also be unpleasant and destructive consequences. Serious caution is required, and the issue should be considered before, rather than after, obvious damage occurs. With the hazards carefully in mind, we propose a framework that should allow a healthy and positive role for speculation
117116150
Exact Renormalization Group for Gauge Theories
Renormalization group ideas have been extremely important to progress in our understanding of gauge field theory. Particularly the idea of asymptotic freedom leads us to hope that nonabelian gauge theories exist in four dimensions and yet are capable of producing the physics we observe — quarks confined in meson and baryon states. For a thorough understanding of the ultraviolet behavior of gauge theories, we need to go beyond the approximation of the theory at some momentum scale by theories with one or a small number of coupling constants. In other words, we need a method of performing exact renormalization group transformations, keeping control of higher order effects, nonlocal effects, and large field effects that are usually ignored.
124187143
On the approach to the critical point
The mass perturbation yields a new construction for Even + Linear quantum field models. For rp4 models, the Schwinger functions are differentiable in the parameter 6 = 5y~, in the single phase region, and are continuous ac. For the Isingd model, d >_ 2 classical bounds on the critical exponents, e. g. 1 y 2v, are a consequence of an Ornstein-Zernike type upper bound on the two point function.
117474017
Absolute bounds on vertices and couplings
We prove absolute upper and/or lower bounds on ~ and cp6 dimensionless vertices and physical coupling constants. 1. DEFINITIONS AND ASSUMPTIONS We derive absolute bounds on ~p4 and qJ6 dimensionless coupling constants g. For instance, in a pure ~p4 model we prove ( 1.1 ) 0 g const., in the single phase region (no symmetry breaking). We also obtain bounds on associated vertex functions and on connected parts. Our general methods presumably have other consequences for n-particle amplitudes, n >_ 8, which we do not pursue here. DEFINITIONS. Let (1) Supported in part by the National Science Foundation under Grant MPS 74-13252. (2) Supported in part by the National Science Foundation under Grant MPS 73-05037. Annales de l’Institut Henri Poincaré Section A Vol. XXII, n° 2 1975. 8 98 J. GLIMM AND A. JAFFE denote the n-point (Euclidean) Schwinger function, and let ( 1 ... n )~ denote its connected part. We let denote the Fourier transform of 1 ... We encounter often, and denote it x(p). We use the standard spectral representation where dp(a) is the spectral measure. Furthermore, we let denote the n-point Euclidean vertex function (thus r(n) is the amputated, one particle irreducible part of ST"~(~)). In an even theory, we have for example, where pi + ... + 7?e = 0, where (X, ~ X) is a partition of 0?i,..., p6) into two subsets of three elements, and where Px is the sum of the momenta in X. We let d 4 denote the space-time dimension and let m denote the mass gap. Note that has the dimension of mass to the power d 2). We define the dimensionless amplitude by and the associated coupling constant by (1.9) = For example, where x * x(0). Annales de l ’Institut Henri Poincaré Section A 99 ABSOLUTE BOUNDS ON VERTICES AND COUPLINGS ASSUMPTIONS. We make some or all of the following assumptions. In the case of weakly coupled assumptions (a)-( f) have been proved, while for single phase, even models (h)-(i) have been proved. We expect (a)-( f ) to hold for single phase and models, while (h)-(i) are expected for all single phase even ~ models, d 4. The evidence for the validity of (1.4) in qJ1 comes from perturbation theory. The proof of (i) in even, single phase has been established by Cartier (private communication). (a) The Wightman axioms for a scalar boson field, or the Osterwalder. Schrader axioms for the corresponding Euclidean theory. The two point function satisfies (1.4). In particular we are assuming that is finite. (b) A positive mass gap, m > 0. M~=0. (d) The second Griffiths inequality, where (Ai, A2) is a partition of (1, ..., n). For instance, ( f ) The measure dp(a) contains a delta function at a = m2, of unit strength (This is a renormalization hypothesis). (h) In the case of a cp4 interaction, it is appropriate to assume the Lebowitz inequality ( 1234 )~ ~ 0, or the set of Lebowitz inequalities, where W is an (odd) subset of U. (i) In the case of a pure cp4 model, we assume ( 1.13) ( 123456 ~T >_ 0. 2. BOUNDS ON QUARTIC COUPLINGS THEOREM 2 .1. (i) Assume (a)-(d) above. Then Vol. XXII, no 2 1975. 100 J. GLIMM AND A. JAFFE (ii) Assume ( f ) also, then x >_ m 2, so (2.2) const. (iii) Assume (h) also, then (2 . 3) 0 g~4~ const. x 2m ~ const. Remark 1. The constants are pure numbers, independent of all parameters. They can be determined explicitly by the proof. Remark 2. The upper bound on is related to the picture of « critical point dominance » discussed in [7]. According to this picture, for d 4, g~4~ assumes its maximum value at the critical point, i. e. at the onset of phase transitions. According to standard ideas, m = 0 at the critical point. Remark 3. For d 4, the limit ~,o ( = bare cp4 coupling constant) oo with m fixed plays a role in the application of the Callan-Symanzik equations to the study of critical phenomena [2]. We use Theorem 2.1 below, combined with bounds from [3], to show that either this limit exists, or that in this limit g(4) 0 (see Corollary 2 . 4). If g~4~ is monotone in the dimensionless charge À-omg-4, then the alternative g(4) ~ 0 is excluded. The possibility that g~4~ is monotone is discussed in [1], and is a consequence of conventional assumptions made in the study of the Callan-Symanzik equations. Remark 4. Other bounds on quartic couplings have been established by Lukaszuk and Martin [4], see also Healy [5]. Their bounds require different assumptions and methods, and yield different conclusions. In the case of a pure cp4 model we also obtain bounds on g~4~(p) 0. THEOREM 2.2. Assume (a)-(h) above. There exists 5 > 0, such that if 6m, then g~4~(p) is analytic in p and bounded uniformly by Proof of Theorem 2.1. By the positivity of dp (Wightman positivity condition) (2. 5) 12~0, so the right side of (1.11) is positive. By symmetry we then have Annales de l’Institut Henri Poincaré Section A 101 ABSOLUTE BOUNDS ON VERTICES AND COUPLINGS Here each Ai is either or else a permutation of (12~(34~13X24). We bound each Ai using Proposition A2.1, of Appendix 2, An elementary estimate shows that each At is integrable over dxidx2dx3. By homogeneity, and (A2.1), from which we conclude By ( 1. 8), with n = 4, and (1. 6) we have (2 .1 ). The theorem then follows. Proof of Theorem 2 . 2. Assuming (h) and (2 . 6), We bound the power series coefficients of expanding about p = 0. The derivatives become multiplication by ix j in the Fourier transform, and are dominated by the exponential decrease of (2. 8), estimated by (A2.1). Thus the power series in momenta converges for I Pi I m/3, and establishes the analyticity and boundedness of St4)(p), Finally, the analyticity and boundedness of follows from (1.4), for I p I m, yielding Thus g(4)(p) is analytic in p and with the constant independent of p, m, A for This completes the proof. We have also proved. COROLLARY 2.3. Assuming (a)-(h), Si-4)(p) is analytic for I ~ ~m and bounded uniformly by (2. 9). We now let be the n-point Schwinger function for a model satisVol. XXII, nO 2 -1975. 102 J. GLIMM AND A. JAFFE fying (a)-(h) and labelled by the index j (for instance, may be the cp2 model, labelled by the bare coupling and mass m). COROLLARY 2.4. Assume mj is bounded away from zero. Then either there exists a convergent subsequence of Schwinger functions i with satisfying (a)-(h), or else 0. Proof Assume for some p, with sm, there is a subsequence of gr(p) bounded away from zero. Then by (2.4) This bound (uniform in r) on proves a uniform bound for a suitable Schwartz space norm j . 109. Thus by Theorem 1 of [3], a convergent subsequence of Schwinger functions exists, and is a tempered distribution. The properties (a)-(h) follow from the corresponding properties in the approximate theories. Conversely, if 0 for each p with |pi| I 6m, the uniform bound (2.4) ensures that the family of analytic functions gj(p) converges to the analytic function zero. 3. BOUNDS ON In this section we prove preliminary bounds on ST. THEOREM 3.1. (i) Assume (a)-(h). Then there exists 6 > 0 such that if then Si:6)(p) is analytic and bounded by (ii) Assume (a)-(i). Then The constants are pure numbers, independent of the parameters. Annales de l’Institut Henri Poincaré Section A 103 ABSOLUTE BOUNDS ON VERTICES AND COUPLINGS Proof. We consider p = 0. The extension to p # 0 follows the proof of Theorem 2.2. Using (/), we have The definition of the six point connected part is Here Y = (Yi, Y2) ranges over the fifteen partitions of (1, 2, ..., 6) into two and four element subsets, while Z = (Zi, Z2, Z3) ranges over the fifteen partitions of (l, ..., 6) into three pairs. Thus where E’ omits the partition ( 12 )( 3456 ). We use (h) and the definition of( 1234 )r to obtain where E’ omits the three partitions which contain the factory Zi) =(12). By symmetry, where j = 1, 2, ..., 15 labels the two element subsets of (1, ... , 6), and (Z~, Z2’~, Z3’~) one of the twelve partitions of (1, ..., 6) into three pairs, no one of which is j. We now follow the proof of Theorem 2.1 to obtain proving one side of (3 .1 ). We next use (1.12) (see [3]) with U = (3456) to obtain where W ranges over the eight odd subsets of U. Thus by (3.4), Vol. XXII, nO 2 1975. 104 J. GLIMM AND A. JAFFE where E" ranges over the six partitions containing both 1 and 2 in the four element subset. By the definition of ( 1234 )T, By (1.11) we note that -.
116890750
Constructive field theory
13103184
An Exchange Identity for Non-linear Fields
We establish a useful identity for intertwining a creation or annihilation operator with the heat kernel of a self-interacting bosonic field theory.
119468667
Gravitational Waves Probe the Coalescence Rate of Massive Black Hole Binaries
We calculate the expected nHz-μHz gravitational wave (GW) spectrum from coalescing massive black hole (MBH) binaries resulting from mergers of their host galaxies. We consider detection of this spectrum by precision pulsar timing and a future pulsar timing array. The spectrum depends on the merger rate of massive galaxies, the demographics of MBHs at low and high redshift, and the dynamics of MBH binaries. We apply recent theoretical and observational work on all of these fronts. The spectrum has a characteristic strain hc(f) ~ 10-15(f/yr-1)-2/3, just below the detection limit from recent analysis of precision pulsar timing measurements. However, the amplitude of the spectrum is still very uncertain owing to approximations in the theoretical formulation of the model, to our lack of knowledge of the merger rate and MBH population at high redshift, and to the dynamical problem of removing enough angular momentum from the MBH binary to reach a GW-dominated regime.
53953960
CMB and Inflation: the Report from Snowmass 2001
Accelerator–based efforts spanning decades have revealed no direct evidence of physics beyond the Standard Model. Although this is likely to change with the Large Hadron Collider (LHC) or possibly even with the current (higher–luminosity) run at the Tevatron, we point out that astrophysical evidence for physics beyond the standard model has already grown to be conclusive. Observations indicate the need for • inflation, • dark energy, • dark matter, and • massive neutrinos. In short, cosmology is delivering on its promise of probing fundamental physics. These results are creating intellectual excitement in the high energy physics community. They also signal an opportunity. The prospects are bright for further progress in our understanding of fundamental physics, via cosmology, and researchers trained in particle physics should not only be interested, but are well–equipped to contribute. Indeed, a number of particle physicists have become actively engaged in astrophysical observations. In this report we review the current status of observations of the temperature of the cosmic microwave background (CMB), and explain how they provide evidence of an inflationary era of expansion in the early Universe and the existence of “dark energy”. One of our intentions in providing this review is to demonstrate that cosmology has a track record which suggests that we have some understanding of the system we are studying. This track record bolsters the case that we will be able to get meaningful answers to the more detailed questions that are motivating planned observations. These future observations, which we discuss, include more detailed measurement of temperature maps, detection and measurement of polarization (including the modes generated by gravitational waves) and Sunyaev–Zeldovich surveys. Of course, observational cosmology is much broader than the topics we are covering, and the interested reader is encouraged to consult the other P4 and E6 subtopic reports.
18640620
Predicting the polarization of the microwave background from the WMAP temperature maps
Abstract In this paper, we study how to predict the polarization of the Cosmic Microwave Background (CMB) using knowledge of only the temperature (intensity) and the cross-correlation between temperature and polarization. We derive a “Wiener prediction” method and apply it to the Wilkinson Microwave Anisotropy Probe (WMAP) all-sky CMB temperature maps and to the MAXIMA field.
15708379
Instrument, method, brightness, and polarization maps from the 2003 flight of BOOMERanG
Aims.We present the BOOMERanG-03 experiment, and the maps of the Stokes parameters I, Q, U of the microwave sky obtained during a 14 day balloon flight in 2003. 
Methods.Using a balloon-borne mm-wave telescope with polarization sensitive bolometers, three regions of the southern sky were surveyed: a deep survey (~90 square degrees) and a shallow survey (~750 square degrees) at high Galactic latitudes (both centered at , Dec ~ −45°) and a survey of ~300 square degrees across the Galactic plane at , dec ~ −47° . All three surveys were carried out in three wide frequency bands centered at 145, 245 and 345 GHz, with an angular resolution of ~ . 
Results.The 145 GHz maps of Stokes I are dominated by Cosmic Microwave Background (CMB) temperature anisotropy, which is mapped with high signal to noise ratio. The measured anisotropy pattern is consistent with the pattern measured in the same region by BOOMERanG-98 and by WMAP. The 145 GHz maps of Stokes Q and U provide a robust statistical detection of polarization of the CMB when subjected to a power spectrum analysis. The amplitude of the detected polarization is consistent with that of the CMB in the CDM cosmological scenario. At 145 GHz, in the CMB surveys, the intensity and polarization of the astrophysical foregrounds are found to be negligible with respect to the cosmological signal. At 245 and 345 GHz we detect ISD emission correlated to the 3000 GHz IRAS/DIRBE maps, and give upper limits for any other non-CMB component. When compared to monitors of different interstellar components, the intensity maps of the surveyed section of the Galactic plane show that a variety of emission mechanisms is present in that region.
43350860
Planck intermediate results. XIV. Dust emission at millimetre wavelengths in the Galactic plane
We use Planck HFI data combined with ancillary radio data to study the emissivity index of the interstellar dust emission in the frequency range 100–353 GHz, or 3–0.8 mm, in the Galactic plane. We analyse the region l = 20 ◦ –44 ◦ and |b |≤ 4 ◦ where the free-free emission can be estimated from radio recombination line data. We fit the spectra at each sky pixel with a modified blackbody model and two opacity spectral indices, βmm and βFIR, below and above 353 GHz, respectively. We find that βmm is smaller than βFIR, and we detect a correlation between this low frequency power-law index and the dust optical depth at 353 GHz, τ353. The opacity spectral index βmm increases from about 1.54 in the more diffuse regions of the Galactic disk, |b| = 3 ◦ –4 ◦ and τ353 ∼ 5 × 10 −5 , to about 1.66 in the densest regions with an optical depth of more than one order of magnitude higher. We associate this correlation with an evolution of the dust emissivity related to the fraction of molecular gas along the line of sight. This translates into βmm ∼ 1.54 for a medium that is mostly atomic and βmm ∼ 1.66 when the medium is dominated by molecular gas. We find that both the two-level system model and magnetic dipole emission by ferromagnetic particles can explain the results. These results improve our understanding of the physics of interstellar dust and lead towards a complete model of the dust spectrum of the Milky Way from far-infrared to millimetre wavelengths.