Reminiscences of My Work with Richard Lewis Arnowitt

Reminiscences of my work with Richard Lewis Arnowitt Pran Nath∗ Department of Physics, Northeastern University, Boston, MA 02115, USA This article contains reminiscences of the collaborative work that Richard Arnowitt and I did together which stretched over many years and encompasses several areas of particle theory. The article is an extended version of my talk at the Memorial Symposium in honor of Richard Arnowitt at Texas A&M, College Station, Texas, September 19-20, 2014. My collaboration with Richard Arnowitt (1928-2014)1 to constrain the parameters of the effective Lagrangian. started soon after I arrived at Northeastern University in The effective Lagrangian was a deduction from the fol- 1966 and continued for many years. In this brief article lowing set of conditions: (i) single particle saturation in on the reminiscences of my work with Dick I review some computation of T-products of currents, (ii) Lorentz in- of the more salient parts of our collaborative work which variance, (iii) “spectator” approximation, and (iv) local- includes effective Lagrangians, current algebra, scale in- ity which implies a smoothness assumption on the ver- variance and its breakdown, the U(1) problem, formu- tices. The above assumptions lead us to the conclusion lation of the first local supersymmetry and the develop- that the simplest way to achieve these constraints is via ment of supergravity grand unification in collaboration an effective Lagrangian which for T-products of three of Ali Chamseddine, and its applications to the search currents requires writing cubic interactions involving π,ρ for supersymmetry. and A1 fields and allowing for no derivatives in the first -order formalism and up to one-derivative in the second order formalism. The effective Lagrangian is to be used I. EFFECTIVE LAGRANGIANS AND to first order in the coupling constants for three point CURRENT ALGEBRA functions [5–7] and to second order in the coupling constants for 4-point functions and to N 2 order in the − In 1964 Gell-Mann [1] proposed that “quark-type” coupling constant for N point functions. The second step equal-time commutation relations for the vector and the consisted of the imposition of the constraints of current axial vector currents of weak interaction theory serve algebra, CVC and PCAC to determine the parameters as a basis for calculations involving strongly interacting appearing in the effective Lagrangian. Thus in addition particles. Combined with the conserved vector current to [5–7] several other applications of the effective La- (CVC), partially conserved vector current (PCAC) and grangian method were made [15–21]. Below we give some the soft pion approximation many successful results were further details of the effective Lagrangian construction. obtained (see, for example, [2]). However, around 1967 We begin by considering a T-product of three currents, an important issue arose which concerned the breakdown i.e. of the soft pion approximation in the analysis of ρ ππ F αµβ < 0 T (Aα(x)V µ(z)Aβ(y)) 0 > , (1) and A ρπ where the soft pion approximation→ gave a c b 1 → ≡ | | very poor results [3, 4]. This problem was overcome For the time ordering x0 > z0 > y0 Eq. (1) can be ex- in work with Dick and Marvin Friedman by giving up panded so that the soft pion approximation and using the effective La- grangian method to compute the mesonic vertices [5–7]. F αµβ = < 0 Aα n><n V µ m><m Aβ 0 > . (2) | a | | c | | b | A number of other techniques were being pursued at that Xn,m time such as Ward identities by Schnitzer and Weinberg [8], phenomenological Lagrangians by Schwinger[9], Wess Here the states n and m can only be either π or A1 and Zumino [10], and by Ben Lee and Nieh [11] and other mesons. It is clear that the matrix elements that are α µ arXiv:1502.00639v1 [physics.hist-ph] 2 Feb 2015 techniques [12–14]. involved are < 0 Aa πq1a1 >, < πq1a1 Vc πq2a2 > and β | | | | Here I describe briefly the approach that Dick < πq2a2 Ab 0 > and additional terms where π is replaced | | 0 0 0 Arnowitt, Marvin Friedman and I followed which first by A1. For another time ordering , i.e., y > x >z one of all involved developing an effective Lagrangian for the has πρA1 system but then using current algebra conditions F αµβ = < 0 Aβ n><n Aα m><m V µ 0 > , (3) | b | | a | | c | Xn,m ∗ Email: [email protected] 1 where the state n must be a π or A state. However, the Obituary: Richard Lewis Arnowitt, Physics Today 67(12), 68 1 (2014). Submitted by Stanley Deser (Brandeis and Caltech), state m must be a ρ state to have non-vanishing matrix Charles Misner (University of Maryland), Pran Nath (Northeast- elements. In addition to the above we must also include ern University), and Marlan Scully (Texas A&M and Princeton two particle intermediate states so that for the time or- University). dering x0 >y0 >z0 we have 2 In a similar way one has β F αµβ = < 0 Aα n,m >< n,m V µ k ><k A 0 > . Aµ(x)= g aµ(x)+ F ∂µφ (x) , (10) | a | | c | | b | a A a π a n,m,kX (4) It should be clear that Eqs. (9) and (10) are a conse- Here, for example, n must be either π or A1, and m quence of single particle saturation assumption and not must be a ρ and k a π or A1. This means that one of the meant to be fundamental postulates. Thus our approach particles in the matrix element < n,m V ν k > must be differs from the one by Lee, Weinberg and Zumino [22]. a “spectator”. Thus one has | | It should now be noted that Eqs. (9) and (10) are to be used in the following way: we solve the Heisenberg equa- µ 3 tions and then use them to first order in the coupling < πq1a1ρ,p1a3 Vc πq2a2 > = δa1a2 δ (~q1 ~q2) constant in the computation of T-product involving three | | µ− < ρp1a3 Vc 0 > . (5) currents. This is equivalent to the in-field expansion of × | | Eq. (7). However, due to the presence of the propagators Such contributions are required by Lorentz invariance µ µ ∆λ etc the locality of Vc (x) etc is not guaranteed. Thus and crossing symmetry. Indeed Eq. (5) is the cross µ µ if we demand that va (x), πa(x) and aa (x) be local field diagram contribution to Eq. (3). Now the vacuum operators, whose commutators for space-like separations to one particle matrix elements of currents define the vanish, then this condition can be guaranteed if we re- interpolating constants Fπ,gA,gρ so that one has < µ α α α ασ quire that the sources for the fields vc etc arise from a 0 Aa (0) πq1a1 >= Fπq , < 0 A (0) A1σ >= gAǫ and | µ | µσ | | Lagrangian with interactions that are cubic in the fields. < 0 V (0) ρσ >= gρǫ (where we have suppressed the Thus the arguments laid out above lead us to an effective normalizations| | and iso-spin factors). These results can µ µ µ µ Lagrangian of the form be simulated by writing Va = gρvã and Aa = gAaã + µ Fπ∂ φã where the tilde fields are the in-fields. For the (3) matrix elements of a current between two particle states, eff = 0 + g 3 (11) µ L L L e.g., < πq1a Vc (0) πq2b> we write (suppressing normal- µ | | µ λ where 0 = 0π + 0ρ + 0A1 . We note that we arrived at ization factors) < πq1a Vc (0) πq2b >= ǫabc∆λΓ (q1, q2) L L L L µ | | λ Eq. (11) purely from the conditions of (i) single particle where ∆λ is a ρ propagator and the vertex Γ can be expanded in a power series saturation of T -products, (ii) spectator approximation, (iii) Lorentz invariance and crossing symmetry and , (iv) locality. The current algebra constraints, i.e., current Γλ(q , q ) = (qλ + qλ)(α + α k2 + ) , (6) algebra commutation relations, CVC and PCAC have 1 2 1 2 1 2 ··· λ λ λ played no role in the analysis thus far. where k = q1 q2 . Now the two particle matrix element µ − µ of V , i.e., < πq1a Vc (0) πq2b >, can be obtained if we µ | | The analysis above can be extended to higher point replace the vc field in terms of the bilinear product of functions. For instance, for the computation of the four two pion in-fields. Including the vacuum to one particle -point function, matrix element contribution this leads us to a following µ αβµν α β µ ν form for Vc . F (x,y,z,w) <T (A (x)A (y)V (z)V (ω)) > , ≡ (12) µ µ 4 µ the implementation of the single particle saturation re- V (x)= gρv˜ + ǫabc d y∆ (x y) c c Z λ − quires that we include the following set of contributions, and (i) diagrams where we have a cascade of three point α α 2 + φ˜ (y)∂λφ˜ (y)+ , (7) × 1 − 2 ··· a b ··· vertices, and (ii) diagrams where we have four point ver- where the dots at the end stand for other bilinear terms tices. The diagrams of type (i) arise from 3, and diagrams of type (ii) arise from . The LagrangianL that are left out. The form above guarantees crossing L4 symmetry. A very similar analysis holds for the axial (4) = + g + g2 , (13) current. For further analysis it is useful to replace the Leff L0 L3 L4 in-fields by the Heisenberg field operators which obey the is to be used to the first non-vanishing order, i.e., order Heisenberg field equations.

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