This letter by G. Parisi discusses the necessity of an infinite number of order parameters in the mean-field approximation of spin-glasses, as described by replica theory. The replica theory introduces an order parameter \( Q_{\alpha, \beta} \) which is the limit of an \( n \times n \) matrix as \( n \) approaches zero. In the mean-field approximation, this parameter is non-zero only in the spin-glass phase at zero magnetic field. The Sherrington-Kirkpatrick (S-K) model, where the mean-field approximation is exact in the thermodynamic limit, is used to illustrate this point. The standard treatment of the S-K model assumes \( Q_{\alpha, \beta} = q \), where \( q \) is the Edwards-Anderson order parameter. However, this assumption leads to discrepancies with computer simulations and a negative entropy at zero temperature. It is suggested that the true value of \( Q_{\alpha, \beta} \) is not symmetric under permutations of indices, breaking replica symmetry. Various patterns of symmetry breaking have been proposed, and the approach has been generalized to matrices parametrized by multiple variables, leading to better agreement with simulations. The author studies the S-K model with matrices parametrized by 1, 3, 5, etc., parameters near the critical temperature \( T_c \). Near \( T_c \), the order parameter \( Q_{\alpha, \beta} \) is small, allowing a Taylor expansion. The free energy \( F(\tau) \) is derived, where \( \tau \) is proportional to \( T_c - T \). The matrix \( Q_{\alpha, \beta} \) is shown to belong to a zero-dimensional space, and the general form of the matrix is discussed. The free energy is maximized with respect to some parameters, a key feature of the replica approach. The consistency of this approach is checked by computing the eigenvalues of the Hessian matrix, which should have at least one zero eigenvalue corresponding to infinite "replicon susceptibility." The physical interpretation of the function \( q(x) \) defined by the parameters \( q_i \) and \( m_i \) is discussed, and the magnetic susceptibility and internal energy are derived. The identification of the Edwards-Anderson order parameter \( q_{\text{ph}} \) is problematic, and the breaking of replica symmetry implies that the Fischer relation may not hold. Recent Monte Carlo simulations suggest that the Fischer relation is not satisfied.This letter by G. Parisi discusses the necessity of an infinite number of order parameters in the mean-field approximation of spin-glasses, as described by replica theory. The replica theory introduces an order parameter \( Q_{\alpha, \beta} \) which is the limit of an \( n \times n \) matrix as \( n \) approaches zero. In the mean-field approximation, this parameter is non-zero only in the spin-glass phase at zero magnetic field. The Sherrington-Kirkpatrick (S-K) model, where the mean-field approximation is exact in the thermodynamic limit, is used to illustrate this point. The standard treatment of the S-K model assumes \( Q_{\alpha, \beta} = q \), where \( q \) is the Edwards-Anderson order parameter. However, this assumption leads to discrepancies with computer simulations and a negative entropy at zero temperature. It is suggested that the true value of \( Q_{\alpha, \beta} \) is not symmetric under permutations of indices, breaking replica symmetry. Various patterns of symmetry breaking have been proposed, and the approach has been generalized to matrices parametrized by multiple variables, leading to better agreement with simulations. The author studies the S-K model with matrices parametrized by 1, 3, 5, etc., parameters near the critical temperature \( T_c \). Near \( T_c \), the order parameter \( Q_{\alpha, \beta} \) is small, allowing a Taylor expansion. The free energy \( F(\tau) \) is derived, where \( \tau \) is proportional to \( T_c - T \). The matrix \( Q_{\alpha, \beta} \) is shown to belong to a zero-dimensional space, and the general form of the matrix is discussed. The free energy is maximized with respect to some parameters, a key feature of the replica approach. The consistency of this approach is checked by computing the eigenvalues of the Hessian matrix, which should have at least one zero eigenvalue corresponding to infinite "replicon susceptibility." The physical interpretation of the function \( q(x) \) defined by the parameters \( q_i \) and \( m_i \) is discussed, and the magnetic susceptibility and internal energy are derived. The identification of the Edwards-Anderson order parameter \( q_{\text{ph}} \) is problematic, and the breaking of replica symmetry implies that the Fischer relation may not hold. Recent Monte Carlo simulations suggest that the Fischer relation is not satisfied.