125224687
Existence of Phase Transitions for phi**4 in Two-Dimensions Quantum Fields
121389988
Whither Axiomatic Field Theory
124867759
The Particle Structure of the Weakly Coupled P(φ)2 Model and Other Applications of High Temperature Expansions: Part I. Physics of Quantum Field Models
116890750
Constructive field theory
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 -.
115879784
Functional Integral Methods in Quantum Field Theory
We give a special form of the Osterwalder-Schrader axioms in terms of conditions on a functional integral S{f} = ∫eiϕ(f)dμ. This yields a simple, self-contained construction of a Hamiltonian H, a relativistic, local boson quantum field Φ, and a Feynman-Kac formula to study perturbations H + Φ of H.
119085401
Imprints of spherical non-trivial topologies on the CMB
The apparent low power in the CMB temperature anisotropy power spectrum derived from WMAP motivated us to consider the possibility of a non-trivial topology. We focus on some simple spherical multi-connected manifolds (Quaternionic, Octahedral, Truncated Cube and Poincare spaces) and discuss their implications for the CMB in terms of the power spectrum, maps and the correlation matrix. We also perform Bayesian model comparison against the fiducial best-fit LCDM based both on the power spectrum and the correlation matrix to assess their statistical significance. We find that the first year power spectrum shows a slight preference for the Truncated Cube space, but the 3-year data show no evidence for any of these spaces.
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.
38675087
DETECTION OF GRAVITATIONAL WAVES FROM INFLATION
Recent measurements of temperature fluctuations in the cosmic microwave background (CMB) indicate that the Universe is flat and that large-scale structure grew via gravitational infall from primordial adiabatic perturbations. Both od these observations seem to indicate that we are on the right track with inflation. But what is the new physics responsible for inflation? This question can be answered with observations of the polarization of the CMB. Inflation predicts robustly the existence of a stochastic background of cosmological gravitational waves with an amplitude proportional to the square of the energy scale of inflation. This gravitational-wave background induces a unique signature in the polarization of the CMB. If inflation took place at an energy scale much smaller than that of grand unification, then the signal will be too small to be detectable. However, if inflation had something to do with grand unification or Planck-scale physics, then the signal is conceivably detectable in the optimistic ...
119098089
Improved method for detecting local discontinuities in CMB data by finite differencing
An unexpected distribution of temperatures in the CMB could be a sign of new physics. In particular, the existence of cosmic defects could be indicated by temperature discontinuities via the Kaiser-Stebbins effect. In this paper, we show how performing finite differences on a CMB map, with the noise regularized in harmonic space, may expose such discontinuities, and we report the results of this process on the 7-year Wilkinson Microwave Anisotropy Probe data.
16998444
The BOOMERanG experiment and the curvature of the Universe
We describe the BOOMERanG experiment and its main result, i.e. the measurement of the large scale curvature of the Universe. BOOMERanG is a balloon-borne microwave telescope with sensitive cryogenic detectors. BOOMERanG has measured the angular distribution of the Cosmic Microwave Background on ∼ 3% of the sky, with a resolution of ∼ 10 arcmin and a sensitivity of ∼ 20μK per pixel. The resulting image is dominated by hot and cold spots with rms fluctuations ∼ 80μK and typical size of ∼ 1o. The detailed angular power spectrum of the image features three peaks and two dips at l = (213−13+10), (541−32+20), (845−25+12) and l = (416−12+22), (750−750+20), respectively. Such very characteristic spectrum can be explained assuming that the detected structures are the result of acoustic oscillations in the primeval plasma. In this framework, the measured pattern constrains the density parameter Ω to be 0.85 < Ω < 1.1 (95% confidence interval). Other cosmological parameters, like the spectral index of initial density fluctuations, the density parameter for baryons, dark matter and dark energy, are detected or constrained by the BOOMERanG measurements and by other recent CMB anisotropy experiments. When combined with other cosmological observations, these results depict a new, consistent, cosmological scenario.