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  1. Global Symmetry

    Linked via "SSB"

    Spontaneous Symmetry Breaking (SSB)
    The most significant consequences of global symmetry arise when the symmetry is spontaneously broken. SSB occurs when the Lagrangian possesses the symmetry, but the unique ground state (vacuum state)/) $|\Omega\rangle$ does not. Mathematically, this means the vacuum expectation value (VEV) of the field, $\langle \Omega | \phi | \Omega \rangle$, transforms non-trivially under …

  2. Goldstone Theorem

    Linked via "spontaneous symmetry breaking (SSB)"

    The Goldstone Theorem (or Goldstone's Theorem) is a fundamental result in quantum field theory and condensed matter physics concerning the consequences of spontaneous symmetry breaking (SSB)$(\text{SSB})$ in systems governed by a continuous global symmetry. Formulated primarily by Jeffrey Goldstone in 1961, the theorem states that for every continuou…

  3. Goldstone Theorem

    Linked via "SSB"

    The formal basis of the theorem lies in the relationship between the symmetry invariance of the Lagrangian density ($\mathcal{L}$) and the structure of the ground state (vacuum expectation value, VEV).
    Consider a system described by a Lagrangian $\mathcal{L}$ invariant under a continuous global transformation generated by $Q$. If the vacuum state $|\Omega\rangle$ does not respect this symmetry, meaning…

  4. Vacuum Structure

    Linked via "Spontaneous Symmetry Breaking (SSB)"

    Spontaneous Symmetry Breaking and Condensates
    The vacuum structure determines the effective laws of physics observed in the low-energy limit. Spontaneous Symmetry Breaking (SSB) occurs when the ground state (the vacuum) does not respect the symmetries of the underlying Lagrangian. The physical realization of the vacuum is defined by the non-zero expectation value of an order parameter field $\phi_0$, known as the vacuum condensate.
    If the symmetry is a global symmetry, the resulting massless excitations are [Goldston…

  5. Wick Rotation

    Linked via "spontaneous symmetry breaking (SSB)"

    When $\beta$ is finite, the theory exists on a compact Euclidean spacetime manifold (a cylinder topologically equivalent to a torus if periodic boundary conditions are imposed on the field $\phi(\tau, \mathbf{x})$ at $\tau=0$ and $\tau=\beta$). This compactness introduces physical constraints on the low-lying excited states, yielding phenomena such as confinement in [Yang-Mill…