118785746
Twist Positivity for Lagrangian Symmetries
We prove twist positivity and positivity of the pair correlation function for combined spatial and internal symmetries of free bosonic Lagrangians. We work in a general setting, extending the results obtained in Twist Positivity (1).
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.
116890750
Constructive 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.
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.
118588305
Likelihood Analysis of Galaxy Surveys
One of the major goals of cosmological observations is to test theories of structure formation. The most straightforward way to carry out such tests is to compute the likelihood function L, the probability of getting the data given the theory. We write down this function for a general galaxy survey. The full likelihood function is very complex, depending on all of the $n$-point functions of the theory under consideration. Even in the simplest case, where only the two point function is non-vanishing (Gaussian perturbations), L cannot be calculated exactly, primarily because of the Poisson nature of the galaxy distribution. Here we expand L about the (trivial) zero correlation limit. As a first application, we take the binned values of the two point function as free parameters and show that L peaks at $(DD - DR + RR)/DD$. Using Monte Carlo techniques, we compare this estimator with the traditional $DD/DR$ and Landy & Szalay estimators. More generally, the success of this expansion should pave the way for further applications of the likelihood function.
17654256
Likelihood analysis of large scale flows
We apply a likelihood analysis to the data of \markcite{Lauer \& Postman 1994} With $P(k)$ parametrized by $(\sigma_8, \Gamma)$, the likelihood function peaks at $\sigma_8\simeq0.3$, $\Gamma\lesssim0.025$, indicating at face value very strong large-scale power, though at a level incompatible with COBE\@. There is, however, a ridge of likelihood such that more conventional power spectra do not seem strongly disfavored. The likelihood calculated using as data only the components of the bulk flow solution peaks at higher $\sigma_8$, in agreement with other analyses, but is rather broad. The likelihood incorporating both bulk flow and shear gives a different picture. The components of the shear are all low, and this pulls the peak to lower amplitudes as a compromise. 
The Lauer \& Postman velocity data alone are therefore {\em consistent}\/ with models with very strong large scale power which generates a large bulk flow, but the small shear (which also probes fairly large scales) requires that the power would have to be at {\em very}\/ large scales, which is strongly disfavored by COBE\@. The velocity data also seem compatible with more conventional $P(k)$ with $0.2\lesssim\Gamma\lesssim0.5$, and the likelihood is peaked around $\sigma_8\sim1$, in which case the bulk flow is a moderate, but not extreme, statistical fluctuation. 
Applying the same techniques to the data of \markcite{Riess, Press, \& Kirshner 1995}, the results are quite different. The flow is not inconsistent with the microwave dipole and we derive only an upper limit to the amplitude of the power spectrum: $\sigma_8\lesssim1.5$ at roughly 99\%.
2204032
Planck 2013 results. II. Low frequency instrument data processing
We describe the processing of the 531 billion raw data samples from the High Frequency Instrument (hereafter HFI), which we performed to produce six temperature maps from the first 473 days of Planck-HFI survey data. These maps provide an accurate rendition of the sky emission at 100, 143, 217, 353, 545, and 857 GHz with an angular resolution ranging from 9.7 to 4.6 arcmin. The detector noise per (effective) beam solid angle is respectively, 10, 6, 12 and 39 microKelvin in HFI four lowest frequency channel (100--353 GHz) and 13 and 14 kJy/sr for the 545 and 857 GHz channels. Using the 143 GHz channel as a reference, these two high frequency channels are intercalibrated within 5% and the 353 GHz relative calibration is at the percent level. The 100 and 217 GHz channels, which together with the 143 GHz channel determine the high-multipole part of the CMB power spectrum (50 < l <2500), are intercalibrated at better than 0.2 %.
13664482
Planck 2013 results. IV. Low Frequency Instrument beams and window functions
This paper presents the characterization of the in-flight beams, the beam window functions and the associated errors for the Planck Low Frequency Instrument (LFI). Knowledge of the beam profiles is the key to determining their imprint on the transfer function from the observed to the actual sky anisotropy power spectrum. The main beam distortions affect the beam window function, complicating the reconstruction of the anisotropy power spectrum at high multipoles, whereas the sidelobes affect the low and intermediate multipoles. The in-flight assessment of the LFI main beams relied on the measurements performed during Jupiter observations. By stacking the data from Jupiter transits, the main beam profiles are measured down to -20 dB at 30 and 44 GHz, and down to -25 dB at 70 GHz. The main beam solid angles are determined to better than 0.2% at each LFI frequency band. To ensure a characterization of the main beam free from the radiometer noise, a dedicated tuning on the Planck pre-launch optical model is performed. This approach provides an optical model whose beams fully reproduce the measurements in the main beam region, but also allow us to describe the beams at power levels lower than can be reached by the Jupiter measurements themselves. The agreement between the simulated beams and the scanning beams is better than 1% at each LFI frequency band. The simulated beams are used for the computation of the window functions for the effective beams. The error budget in the window functions was estimated considering both main beam and sidelobe contributions, as well as taking into account the radiometer bandshapes. The total uncertainties in the effective beam window functions are: (at ell = 600) 2% and 1.2% at 30 and 44 GHz, respectively; and at ell = 1000, 0.7% at 70 GHz.
15450342
THE ANGULAR THREE-POINT CORRELATION FUNCTION IN THE QUASI-LINEAR REGIME
We calculate the normalized angular three-point correlation function (3PCF), q, as well as the normalized angular skewness, s{sub 3}, assuming the small-angle approximation, for a biased mass distribution in flat and open cold dark matter (CDM) models with Gaussian initial conditions. The leading-order perturbative results incorporate the explicit dependence on the cosmological parameters, the shape of the CDM transfer function, the linear evolution of the power spectrum, the form of the assumed redshift distribution function, and linear and nonlinear biasing, which may be evolving. Results are presented for different redshift distributions, including that appropriate for the APM Galaxy Survey, as well as for a survey with a mean redshift of z{approx_equal}1 (such as the VLA FIRST Survey). Qualitatively, many of the results found for s{sub 3} and q are similar to those obtained in a related treatment of the spatial skewness and 3PCF, such as a leading-order correction to the standard result for s{sub 3} in the case of nonlinear bias (as defined for unsmoothed density fields), and the sensitivity of the configuration dependence of q to both cosmological and biasing models. We show that since angular correlation functions (CFs) are sensitive to clustering over a range of redshifts, themore » various evolutionary dependences included in our predictions imply that measurements of q in a deep survey might better discriminate between models with different histories, such as evolving versus nonevolving bias, that can have similar spatial CFs at low redshift. Our calculations employ a derived equation, valid for open, closed, and flat models, to obtain the angular bispectrum from the spatial bispectrum in the small-angle approximation. (c) (c) 2000. The American Astronomical Society.« less