119588441
Response to comments on ``Theoretical Mathematics''
The authors discuss various objections and rejoinders in the collected responses [math.HO/9404229,math.HO/9404236] to their original article on the relationship between mathematics and theoretical physics [math.HO/9307227].
116890750
Constructive field theory
125224687
Existence of Phase Transitions for phi**4 in Two-Dimensions Quantum Fields
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 -.
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.
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.
36839918
A Measurement of the Polarization-Temperature Angular Cross-Power Spectrum of the Cosmic Microwave Background from the 2003 Flight of BOOMERANG
We present a measurement of the polarization-temperature angular cross power spectra, langTErang and langTBrang, of the cosmic microwave background. The result is based on ~200 hr of data from eight polarization-sensitive bolometers operating at 145 GHz during the 2003 flight of BOOMERANG. We detect a significant langTErang correlation in the l-range between 50 and 950 with a statistical significance of >3.5 σ. Contamination by polarized foreground emission and systematic effects are negligible in comparison with statistical uncertainties. The spectrum is consistent with previous detections and with the "concordance model" that assumes adiabatic initial conditions. This is the first measurement of polarization-temperature angular cross-power spectra using bolometric detectors.
681194
Observational Constraints on Microwave Anisotropy from Point Sources
Applying basic physical principles to recent observational results, we derive upper and lower limits on microwave anisotropy from point sources over the range of frequencies 10-1000 GHz. We examine the level of noise in the observations as a possible indication of source confusion at subarcminute scales. We also derive an upper limit on microwave anisotropy caused by the sources responsible for the Far-Infrared Background radiation detected in FIRAS data. Our upper limit on point source confusion of DeltaT/T=10^{-5} for a 10' beam at 100 GHz would cause severe foreground contamination for CMB anisotropy observations, although the actual contamination level is probably much lower. This upper limit constrains the long-feared possibility of an undetected population of sources with emission peaking near 100 GHz. Source detections closer to 100 GHz are needed to improve our knowledge of galaxy evolution at high redshift and to predict the level of point source confusion.
16086049
Data Compression for CMB Experiments
We discuss data compression for CMB experiments. Although "radical compression" to C_l bands, via quadratic estimators or local bandpowers, potentially offers a great savings in computation time, they do a considerably worse job at recovering the full likelihood than the the signal-to-noise eigenmode method of compression.
1392831
Nonlinear Effects in the Amplitude of Cosmological Density Fluctuations
The amplitude of cosmological density fluctuations, sigma_8, has been studied and estimated by analysing many cosmological observations. The values of the estimates vary considerably between the various probes. However, different estimators probe the value of sigma_8 in different cosmological scales and do not take into account the nonlinear evolution of the parameter at late times. We show that estimates of the amplitude of cosmological density fluctuations derived from cosmic flows are systematically higher than those inferred at early epochs from the CMB because of nonlinear evolution at later times. We discuss the past and future evolution of linear and nonlinear perturbations, derive corrections to the value of sigma_8 and compare amplitudes after accounting for these differences.
16795069
A measurement of the angular power spectrum of the CMB temperature anisotropy from the 2003 flight of Boomerang
We report on observations of the cosmic microwave background (CMB) obtained during the 2003 January flight of BOOMERANG. These results are derived from 195 hr of observation with four 145 GHz polarization-sensitive bolometer (PSB) pairs, identical in design to the four 143 GHz Planck High Frequency Instrument (HFI) polarized pixels. The data include 75 hr of observations distributed over 1.84% of the sky with an additional 120 hr concentrated on the central portion of the field, which represents 0.22% of the full sky. From these data we derive an estimate of the angular power spectrum of temperature fluctuations of the CMB in 24 bands over the multipole range 50 ≤ l ≤ 1500. A series of features, consistent with those expected from acoustic oscillations in the primordial photon-baryon fluid, are clearly evident in the power spectrum, as is the exponential damping of power on scales smaller than the photon mean free path at the epoch of last scattering (l ≳ 900). As a consistency check, the collaboration has performed two fully independent analyses of the time-ordered data, which are found to be in excellent agreement.