2  Axiomatic framework and QCD

2.1 Wightman axioms

Wightman axioms need to be thought of as a minimal set of properties that any Quantum Field Theory (QFT) should satisfy, without any reference to perturbation theory or the existence of a Lagrangian. This conceptual framework is particularly useful in discussing properties of Quantum Chromodynamics (QCD) at the non-perturbative level. One needs to keep in mind that concrete QFTs may satisfy extra properties, which are not captured by the Wightman axioms.

  1. As for any Quantum Mechanical system, the set of normalizable states is a separable Hilbert space \(\mathcal{H}\).

    • Given two states \(\Phi\) and \(\Psi\) then any linear combination \(\alpha \Phi + \beta \Psi\) is also a state, where \(\alpha\) and \(\beta\) are complex numbers. Therefore, the set of states is a complex vector space.

    • A scalar product \((\Phi,\Psi)\) is defined on the set of states. The scalar product has the following properties:

      • \((\Phi,\Psi)\) is a complex number;
      • \((\Xi, \alpha \Phi + \beta \Psi) = \alpha (\Xi,\Phi) + \beta (\Xi,\Psi)\) [linerarity in the second argument];
      • \((\Phi,\Psi) = (\Psi,\Phi)^*\) [hermiticity];
      • \((\Phi,\Phi) > 0\) if \(\Phi \neq 0\) [positivity].

    • A finite or countable orthonormal basis \(e_{n=1,2,3\dots}\) exists, such that every state can be written as: \[ \begin{gather*} \Phi = \sum_n c_n e_n \ , \end{gather*} \] where \(c_n\) are complex numbers. This property is called separability. Recall that orthonormality means that: \[ \begin{gather*} (e_n,e_m) = \delta_{nm} \ . \end{gather*} \]

    • Any expression of the type \[ \begin{gather*} \sum_n c_n e_n \qquad \text{with } \sum_n |c_n|^2 < \infty \end{gather*} \] is a state. This property is called completeness.

  2. A unitary representation of the Poincaré group is defined on the Hilbert space of states.

    Elements of the Poincaré group are denoted by \((\Lambda, a)\) and are obtained by composing a proper orthochronous Lorentz transformation \(\Lambda\) with a spacetime translation \(a\). The element \((\Lambda, a)\) acts on spacetime points as: \[ \begin{gather*} x \to x' = \Lambda x + a \ . \end{gather*} \tag{2.1}\] Acting with two elements of the Poincaré group, \((\Lambda_1, a_1)\) first and \((\Lambda_2, a_2)\) after, gives: \[ \begin{gather*} x \to x' = \Lambda_1 x + a_1 \to x'' = \Lambda_2 x' + a_2 = (\Lambda_2 \Lambda_1) x + (\Lambda_2 a_1 + a_2) \ . \end{gather*} \tag{2.2}\] Therefore, the composition law of the Poincaré group reads: \[ \begin{gather*} (\Lambda_2, a_2) \circ (\Lambda_1, a_1) = (\Lambda_2 \Lambda_1, a_2 + \Lambda_2 a_1) \ . \end{gather*} \tag{2.3}\] A unitary representation of the Poincaré group is defined by specifying unitary operators \(U(\Lambda, a)\) acting on the Hilbert space of states such that: \[ \begin{gather*} U(\Lambda_2, a_2) U(\Lambda_1, a_1) = U(\Lambda_2 \Lambda_1, a_2 + \Lambda_2 a_1) \ . \end{gather*} \tag{2.4}\] Using the composition law and by defining \(U(\Lambda) = U(\Lambda, 0)\) and \(U(a) = U(1, a)\), we can write any Poincaré transformation as the composition of a Lorentz transformation and a translation: \[ \begin{gather*} U(\Lambda, a) = U(a) U(\Lambda) \ . \end{gather*} \tag{2.5}\]

    The generator of translations is the energy-momentum operators \(P_\mu\). In terms of the generator, the translation operator reads: \[ \begin{gather*} U(a) = e^{i P a} \ . \end{gather*} \tag{2.6}\]

    In order to be able to describe fermionic particles, one needs to replace the proper orthochronous Lorentz group \(SO^+(1,3)\) with its universal covering group, the set of \(2 \times 2\) complex matrices with determinant equal to 1, denoted by \(\text{SL}(2,\mathbb{C})\). To each matrix \(V \in \text{SL}(2,\mathbb{C})\) one can associate a proper orthochronous Lorentz transformation \(\Lambda(V)\) in the following way.

    Let \(\sigma_k\) for \(k=0,1,2,3\) be the Pauli matrices and let \(\sigma_0\) be the identity matrix. Given a spacetime point \(x \in \mathbb{R}^4\), one can construct a \(2 \times 2\) hermitian matrix \(x^\mu \sigma_\mu\). Notice that \[ \begin{gather*} \det (x^\mu \sigma_\mu) = \det \begin{pmatrix} x^0 + i x^3 & x^1 - i x^2 \\ x^1 + i x^2 & x^0 - i x^3 \end{pmatrix} = x^2 \ . \end{gather*} \] Given \(V \in \text{SL}(2,\mathbb{C})\), notice that the matrix \(V \sigma_\mu V^\dag\) is also hermitian. Since \(\sigma_\mu\) is a basis of the space of \(2 \times 2\) hermitian matrices, we can write: \[ \begin{gather*} V (x^\mu \sigma_\mu) V^\dag = (x')^\mu \sigma_\mu \ , \end{gather*} \] for some \(x' \in \mathbb{R}^4\). Taking the determinant of both sides and using the fact that \(\det V = 1\), we find: \[ \begin{gather*} x^2 = \det (x^\mu \sigma_\mu) = \det [ V^\dag (x^\mu \sigma_\mu) V ] = \det (x'^\mu \sigma_\mu) = (x')^2 \ . \end{gather*} \] Therefore \(x'\) and \(x\) are related by a Lorentz transformation, which we denote as \(\Lambda(V)\): \[ \begin{gather*} x'^\mu = \Lambda^\mu{}_\nu (V) x^\nu \ . \end{gather*} \] Using the orthogonality of the matrices \(\sigma_\mu\), we can also write an explicit expression for the Lorentz transformation associated to \(V\): \[ \begin{gather*} \Lambda^\mu{}_\nu (V) = \frac{1}{2} \tr ( \bar{\sigma}^\mu V \sigma_\nu V^\dag ) \ , \end{gather*} \] where, again, \(\bar{\sigma}^k\) are the Pauli matrices for \(k=1,2,3\) and \(\bar{\sigma}^0\) is the identity matrix.

    1. The Lorentz transformation \(\Lambda(V)\) is uniquely defined by the equation \[ \begin{gather*} V \sigma_\nu V^\dag = \Lambda^\mu{}_\nu (V)\sigma_\mu \ . \end{gather*} \] It follow that \[ \begin{gather*} \Lambda^\mu{}_\nu(V_1V_2) \sigma_\mu = V_1 V_2 \sigma_\nu V_2^\dag V_1^\dag = \Lambda^\rho{}_\nu(V_2) V_1 \sigma_\rho V_1^\dag = \Lambda^\rho{}_\nu(V_2) \Lambda^\mu{}_\rho(V_1) \sigma_\mu \ , \end{gather*} \] which implies \[ \begin{gather*} \Lambda(V_1 V_2) = \Lambda(V_1) \Lambda(V_2) \ . \end{gather*} \] In other words, the map \(V \mapsto \Lambda(V)\) preserves the group structure.

    2. A generic element \(V\) of \(\text{SL}(2,\mathbb{C})\) can be written as \(V = v^\mu \sigma_\mu\), where \(v \in \mathbb{C}^4\) satisfies \(\det V = v^2 = 1\). When continuously shrinking \(\vec{v}\) to zero, \(v^0\) moves with continuity either to \(+1\) or to \(-1\). Therefore, the matrix \(V\) moves with continuity to either the identity matrix \(I\) or to the \(-I\). However \(I\) and \(-I\) are connected by the path \(\text{diag}(e^{i\alpha}, e^{-i\alpha}) \in \text{SL}(2,\mathbb{C})\) for \(\alpha \in [0,\pi]\). It follows that \(\text{SL}(2,\mathbb{C})\) is connected. Since the map \(V \mapsto \Lambda(V)\) is continuous, it follows that its image is contained in the connected component of the identity of the Lorentz group, i.e. the proper orthochronous Lorentz group \(SO^+(1,3)\).

    3. In fact, the image of the map \(V \mapsto \Lambda(V)\) is precisely the proper orthochronous Lorentz group \(SO^+(1,3)\). This can be seen by explicit construction. Every proper orthochronous Lorentz transformation can be written as a composition of a boost and a rotation. A boost with rapidity \(\eta \in \mathbb{R}\) in the direction defined by \(\vec{n} \in \mathbb{R}^3\) with \(\vec{n}^2 = 1\) is represented by the matrix: \[ \begin{gather*} V = \sigma_0 \, \cosh \frac{\eta}{2} + \vec{n} \cdot \vec{\sigma} \, \sinh \frac{\eta}{2} \ , \end{gather*} \] while a rotation by an angle \(\theta \in \mathbb{R}\) around the direction defined by \(\vec{n} \in \mathbb{R}^3\) with \(\vec{n}^2 = 1\) is represented by the matrix: \[ \begin{gather*} V = \sigma_0 \, \cos \frac{\theta}{2} - i \vec{n} \cdot \vec{\sigma} \, \sin \frac{\theta}{2} \ . \end{gather*} \]

    4. From the definition, one easily sees that \(\Lambda(V) = \Lambda(-V)\), i.e. \(V\) and \(-V\) are mapped to the same Lorentz transformation. In fact, using the representation \(V = v^\mu \sigma_\mu\) with \(v^2=1\) and the algebra of the Pauli matrices, one can easily check that \(\Lambda(V_1) = \Lambda(V_2)\) if and only if \(V_1 = \pm V_2\). This means that the map \(V \mapsto \Lambda(V)\) is a two-to-one map from \(\text{SL}(2,\mathbb{C})\) to the proper orthochronous Lorentz group \(SO^+(1,3)\). In other words, \(SL(2,\mathbb{C})\) is a double covering of the proper orthochronous Lorentz group \(SO^+(1,3)\).

    The double covering of the Poincaré group is then given by the set of pairs \((V, a)\), where \(V \in \text{SL}(2,\mathbb{C})\) and \(a \in \mathbb{R}^4\). The action of the double covering of the Poincaré group on spacetime points is given by: \[ \begin{gather*} x \to x' = \Lambda(V) x + a \ . \end{gather*} \] The composition law of the double covering of the Poincaré group reads: \[ \begin{gather*} (V_2, a_2) \circ (V_1, a_1) = (V_2 V_1, a_2 + \Lambda(V_2) a_1) \ . \end{gather*} \] In order to be able to describe fermionic particles, one needs to require that the Hilbert space of states is a unitary representation of the double covering of the Poincaré group. In this case, the unitary operators \(U(V, a)\) acting on the Hilbert space of states satisfy: \[ \begin{gather*} U(V_2, a_2) U(V_1, a_1) = U(V_2 V_1, a_2 + \Lambda(V_2) a_1) \ . \end{gather*} \]

  3. The spectrum of the energy-momentum operator \(P_\mu\) is contained in the closed forward light cone, i.e. it satisfies: \[ \begin{gather*} \sigma(P) \subseteq \overline{\Lambda^+} = \{ p \in \mathbb{R}^4 \text{ s.t. } p^2 \ge 0 \text{ and } p_0 \ge 0 \} \ . \end{gather*} \tag{2.7}\] In other words, no state with negative energy exists and no state with non-vanishing space-like four-momentum exists.

  4. There is a unique normalized state \(| \Omega \rangle\) (unique up to a phase, of course), called the vacuum, which has zero energy-momentum, i.e. \(P_\mu | \Omega \rangle = 0\). Equivalently, the vacuum is the unique state which is invariant under the action of translations, i.e. \(U(a) | \Omega \rangle = | \Omega \rangle\) for all \(a \in \mathbb{R}^4\).

  5. Field operators exist, which we will denote generically as \(\phi(x)\). Keep in mind that there can be several of such fields and that they can carry indices (e.g. vector or spinor indices or indices corresponding to internal symmetries). These fields are not proper operators acting on the Hilbert space, but rather operator-valued distributions. We will still refer to the as operators, as per common practice.

    A formal definition of distributions and operator-valued distributions is beyond the scope of these lectures, but one can think of them as mathematical objects that are defined by their action on test functions (see here for a review of main facts and ideas about distributions). Objects like the Dirac delta \(\delta(x)\) or the Cauchy principal value of \(1/x\) are examples of distributions, and they are defined by means of their integral against (or, action on) test functions \(f(x)\): \[ \begin{align*} & \int dx \, f(x) \delta(x) \equiv f(0) \ , \\ & \int dx \, f(x) \frac{\text{P}}{x} \equiv \lim_{\epsilon \to 0^+} \int_{|x| > \epsilon} dx \, \frac{f(x)}{x} = \int_0^\infty dx \, \frac{f(x) - f(-x)}{x} \ . \end{align*} \tag{2.8}\] In the same way, the field operators \(\phi(x)\) are defined by their action on test functions \(f(x)\): \[ \begin{gather*} \phi(f) = \int d^4 x \, f(x) \phi(x) \ . \end{gather*} \tag{2.9}\] The integrated field operator \(\phi(f)\) is a proper (generally unbounded) operator defined on some dense subspace of the Hilbert space of states, for every Schwartz function \(f(x)\). We recall that Schwartz functions are smooth functions that decrease faster than any inverse polynomial at infinity, together with all their derivatives. We will refer to \(\phi(f)\) as an integrated field operator.

    It is useful to introduce also the related concept of smeared field operator \(\phi_f(x)\), which is defined as the convolution of the field operator with a test function, i.e. \[ \begin{gather*} \phi^f(x) = \int d^4 y \, f(x-y) \phi(y) \ . \end{gather*} \tag{2.10}\] Notice that a smeared field is a particular case of an integrated field, i.e. \(\phi^f(x) = \phi(f_x)\), where the family of functions \(y \mapsto f_x(y)\) parametrized by \(x\) is defined by \(f_x(y) = f(x-y)\).

    One may argue that integrated fields are, in a sense, more physical than the local ones. Consider an experiment that prepares a state \(| \Psi \rangle\) at time zero. The state evolves in time and, at time \(t\), a detector measures the value of the energy density at a point \(\vec{x}\) in space. The outcome is probabilistic and the expectation value of this measurement is given by \(\langle \Psi | T_{00}(t,\vec{x}) | \Psi \rangle\), where \(T_{\mu\nu}(t,\vec{x})\) is the energy-momentum tensor at time \(t\) and position \(\vec{x}\). However, this setup is highly ideal. In practice, the detector does not measure the energy density at a point in space and at a single instant in time: it rather measures the energy in small region of space and integrated over a small time interval. A more realistic detector is described by a function \(f(z)\) which is localized in space and time around the point \((t,\vec{x})\) and which describes the efficiency of the detector in measuring the energy density at the point \(z\). The outcome of the measurement is then given by the expectation value \(\langle \Psi | T_{00}(f) | \Psi \rangle\), where \(T_{\mu\nu}(f)\) is the integrated energy-momentum tensor with test function \(f(z)\).

  6. Fields transform locally under the action of the Poincaré group, i.e. \[ \begin{gather*} U(\Lambda, a) \phi_n(x) U(\Lambda, a)^{-1} = R_{nm}(\Lambda^{-1}) \phi_m (\Lambda x + a) \ , \end{gather*} \tag{2.11}\] where \(R(\Lambda)\) is a finite-dimensional representation of the Lorentz group which acts on the indices of the field \(\phi(x)\). For instance, scalar and vector fields transform as \[ \begin{align*} & U(\Lambda, a) \phi(x) U(\Lambda, a)^{-1} = \phi(\Lambda x + a) \ , \\ & U(\Lambda, a) V_\mu(x) U(\Lambda, a)^{-1} = \Lambda^\nu{}_\mu V_\nu (\Lambda x + a) \ . \end{align*} \tag{2.12}\]

    In particular, notice that this implies \[ \begin{gather*} \phi_m (x + a) = e^{i P a} \phi_m(x) e^{-i P a} \ , \end{gather*} \tag{2.13}\] i.e. the field operators \(\phi_m(x)\) need to be thought as operators in the Heisenberg picture.

    In order to include fermionic fields, one needs to require that the field operators \(\phi_n(x)\) transform locally under the double covering of the Poincaré group, i.e. \[ \begin{gather*} U(V, a) \phi_n(x) U(V, a)^{-1} = R_{nm}(V^{-1}) \phi_m (\Lambda(V) x + a) \ , \end{gather*} \] where \(R(V)\) is a finite-dimensional representation of \(SL(2,\mathbb{C})\), the double-covering of the proper orthochronous Lorentz group.

  7. Field operators commute or anticommute at space-like separations, i.e. if \((x-y)^2 < 0\), then \[ \begin{align*} & [\phi(x), \phi'(y)] = 0 \ , \\ & \{\psi(x), \psi'(y)\} = 0 \ , \\ & [\phi(x), \psi(y)] = 0 \ , \end{align*} \tag{2.14}\] where \(\phi(x)\) and \(\phi'(y)\) are bosonic fields, \(\psi(x)\) and \(\psi'(y)\) are fermionic fields. This property is called microcausality.

    Consider two physical observables (therefore bosonic operators) \(A\) and \(B\), which depend on the fields only in the spacetime regions \(D_A\) and \(D_B\), respectively. If the regions \(D_A\) and \(D_B\) are space-like separated, the time ordering of \(A\) and \(B\) depends on the reference frame. Therefore, the outcome of the measurement of \(A\) and \(B\) must not depend on the order in which the measurements are performed. This implies that \(A\) and \(B\) must commute, i.e. \([A,B] = 0\) (see Problem 2.3). Since this property holds for any pair of observables, it is natural to require that it holds at the level of the fields.

  8. Any normalizable state can be approximated arbitrarily well by applying polynomials of the smeared fields \(\phi_n(f)\) to the vacuum \(| \Omega \rangle\). In more intuitive terms, the set of fields is large enough to capture all the degrees of freedom of the theory and any state is essentially obtained by perturbing the vacuum with the action of fields. Moreover the following technical property is required: states constructed by applying polynomials of the smeared fields to the vacuum belong to the domain of the

2.2 QFTs with a mass gap

In addition to the Wightman axioms, we will require the following properties, which captures the essential features of a QFT with a mass gap, such as QCD.

  1. The spectrum of the invariant mass operator \(M = \sqrt{P_\mu P^\mu}\) contains the zero eigenvalue, which is non-degenerate and corresponds to the vacuum state \(| \Omega \rangle\) and an isolated eigenvalue \(m > 0\). The remaining part of the spectrum is contained in \([m + \Delta, \infty)\) for some \(\Delta > 0\). The spectrum is empty in the interval \((0,m)\) and \((m,m+\Delta)\): these intervals are commonly referred to as mass gaps.

    Let \(\mathcal{H}_m^{1P}\) be the eigenspace of \(M\) corresponding to the eigenvalue \(m\). Since the operator \(M\) is Poincaré invariant, then \(\mathcal{H}_m^{1P}\) is closed under the action of the Poincaré group and can be decomposed into irreducible representations \(\mathcal{H}_{m,i}^{1P}\): \[ \begin{gather*} \mathcal{H}_m^{1P} = \bigoplus_{i=1}^N \mathcal{H}_{m,i}^{1P} \ . \end{gather*} \tag{2.15}\] The spaces \(\mathcal{H}_{m,i}^{1P}\) are one-particle Hilbert spaces with mass \(m\), where the index \(i\) labels different particle species. In the following, we will assume that the number of particle species is finite, i.e. \(N < \infty\). While the different particles have the same mass, they can have different spins \(s_i\) and internal quantum numbers. In practice, this means that a generic state \(\Psi\) in \(\mathcal{H}_m^{1P}\) can be written as \[ \begin{gather*} | \Psi \rangle = \sum_{i=1}^N \sum_{r=1}^{2s_i+1} \int \frac{d^3 p}{(2\pi)^3 2 E(\vec{p})} \, f_{r,i}(\vec{p}) | \vec{p}, r, i \rangle \ . \end{gather*} \tag{2.16}\] Here, \(f_{r,i}(\vec{p})\) is the wave-function in momentum space. \(| \vec{p}, r, i \rangle\) is a one-particle state with momentum \(\vec{p}\), polarization \(r\), and particle species \(i\), normalized as \[ \begin{gather*} \langle \vec{p}', r', i' | \vec{p}, r, i \rangle = 2 E(\vec{p}) (2\pi)^3 \delta^3(\vec{p} - \vec{p}') \delta_{rr'} \delta_{ii'} \ , \end{gather*} \tag{2.17}\] and, finally, the energy of a particle with mass \(m\) and three-momentum \(\vec{p}\) is denoted by \(E(\vec{p}) = \sqrt{m^2 + \vec{p}^2}\). The action of the Poincaré group on one-particle states is defined by \[ \begin{align*} & P_\mu | \vec{p}, r, i \rangle = p_\mu | \vec{p}, r, i \rangle \bigg|_{p_0 = E(\vec{p})} \ , \\ & U(\Lambda) | \vec{p}, r, i \rangle = \sum_{r'} \mathcal{R}_{rr'}^i(\Lambda,\vec{p}) | \vec{p}_\Lambda, r', i \rangle\bigg|_{(E(\vec{p}_\Lambda),\vec{p}_\Lambda) = \Lambda (E(\vec{p}),\vec{p})} \ . \end{align*} \tag{2.18}\]

    The states with invariant mass not smaller than \(m + \Delta\) may be multi-particle states, or one-particle states corresponding to particles with mass not smaller than \(m + \Delta\), or even states that cannot be understood in terms of particles.

  2. If \(| \vec{p}, r, i \rangle\) is a one-particle state with momentum \(\vec{p}\), polarization \(r\), and particle species \(i\), then a field \(\phi_i(x)\) exists such that \(\langle \vec{p}, r, i | \phi_i(x)^\dag | \Omega \rangle\) is not identically zero for every \(x\) and \(r\). The field \(\phi_i^\dag(x)\) is called an interpolating field for the particle with species \(i\).

2.3 The case of QCD

We want to discuss how some of the general concepts introduced above specialize in the case of Quantum Chromodynamics (QCD). For simplicity, we will consider the so-called two-flavour QCD: the gluon field \(A_\mu(x)\) is coupled to two quark fields, the up and down quark fields, which we will denote as \(u(x)\) and \(d(x)\), or alternatively as \(\psi_f(x)\) with \(f = u,d\). We will consider the up and down quarks to have the same mass. While several explicit (perturbative) constructions of the QCD Hilbert space go though some larger vector space which contains unphysical states, such as ghosts and states with zero and negative norm, eventually one needs to project out these unphysical states in order to obtain a physical Hilbert space. The physical Hilbert space of QCD contains only gauge-invariant states which have positive norm. This is the Hilbert space that one should consider in the axiomatic framework.

The fundamental field of QCD, i.e. the gluon field \(A_\mu(x)\) and the quark fields \(\psi_f(x)\), are not invariant under gauge transformations and, in particular, cannot be thought as operators on the physical Hilbert space: when acting on states of the physical Hilbert space they give states that are not gauge invariant. The fields which appear in Wightman axioms must be identified in this case with composite gauge-invariant local functions of the fundamental fields. Examples are: \[ \begin{gather*} \tr F_{\mu\nu}(x) F^{\mu\nu}(x) \ , \qquad \bar{u}(x) \gamma_5 d(x) \ , \qquad \bar{u}(x) \gamma_\mu u(x) + \bar{d}(x) \gamma_\mu d(x) \ , \end{gather*} \tag{2.19}\] or Noether currents such as the energy-momentum tensor \(T_{\mu\nu}(x)\). To be precise, since we are thinking about QCD without a UV regulator, these composite fields should always be understood as properly renormalized. As the issue of renormalization is not particularly relevant for this course, we will ignore it altogether and we will not discuss it further.

The lightest particles produced by QCD are the three pions \(\pi^0\), \(\pi^+\), and \(\pi^-\), which are particles with spin 0, negative intrinsic parity and identical mass \(m_\pi \approx 140 \text{ MeV}\). At the best of our knowledge, the spectrum of the invariant mass operator \(M\) is given by: \[ \begin{gather*} \sigma(M) = \{ 0 \} \cup \{ m_\pi \} \cup [2 m_\pi , \infty) \ , \end{gather*} \tag{2.20}\] and it is consistent with the general form considered in the previous section.

Interpolating fields for the pions can be constructed by considering the internal symmetries of QCD and the way in which the pions transform under these symmetries (we will not discuss this further here): \[ \begin{align*} & \pi_0(x) = \bar{u}(x) \gamma_5 u(x) - \bar{d}(x) \gamma_5 d(x) \ , \\ & \pi_+(x) = \bar{u}(x) \gamma_5 d(x) \ , \\ & \pi_-(x) = \bar{d}(x) \gamma_5 u(x) \ . \end{align*} \tag{2.21}\]

Besides pions, QCD with two flavours is expected to describe other particles, such as protons and neutrons whose mass is approximately \(m_N \approx 940 \text{ MeV}\), but also heavier nuclei such as deuterium, tritium, helium, and so on. Since baryon number is conserved, one can split the Hilbert space of QCD into sectors with fixed baryon number \(B\), i.e. \[ \begin{gather*} \mathcal{H} = \bigoplus_{B=-\infty}^\infty \mathcal{H}_{B} \ . \end{gather*} \tag{2.22}\] Then one can look at the spectrum \(\sigma_B(M)\) of the invariant mass operator \(M\) restricted to the sector with baryon number \(B\). One expects the following structure: \[ \begin{align*} & \sigma_0(M) = \{ 0 \} \cup \{ m_\pi \} \cup [2 m_\pi , \infty) \ , \\ & \sigma_{\pm 1}(M) = \{ m_N \} \cup [m_N + m_\pi , \infty) \ , \\ & \sigma_{\pm 2}(M) = \{ m_D \} \cup [m_D + m_\pi , \infty) \ \dots \end{align*} \tag{2.23}\] where \(m_N\) is the mass of the nucleon and \(m_D\) is the mass of the deuteron.

2.4 Wightman functions

When studying QFT at the perturbative level, one is primarily interested in time-ordered n-point functions, e.g. \[ \begin{gather*} \langle \Omega | \text{T} \{ \phi(x_1) \phi(x_2) \cdots \phi(x_n) \} | \Omega \rangle \ , \end{gather*} \tag{2.24}\] where \(\text{T}\) denotes that the fields must be sorted in the product in such a way that the time arguments \(x_i^0\) are ordered from the largest to the smallest. In the axiomatic framework, one is primarily interested in Wightman functions which are defined as the vacuum expectation value of the product of fields without any time ordering: \[ \begin{gather*} \langle \Omega | \phi(x_1) \phi(x_2) \cdots \phi(x_n) | \Omega \rangle \ . \end{gather*} \tag{2.25}\]

We want to point out that both time-ordered n-point functions and Wightman functions are not functions, in spite of their name, but rather tempered distributions. In the case of the free scalar theory, the time-ordered 2-point function is nothing but the Feynman propagator and reads \[ \begin{align*} \langle \Omega | \text{T} \{ \phi(x) \phi(0) \} | \Omega \rangle = & \lim_{\epsilon \to 0^+} \int \frac{d^4 p}{(2\pi)^4} \, \frac{i \, e^{-i p x}}{p^2 - m^2 + i \epsilon} \nonumber \\ = & \lim_{\epsilon \to 0^+} \frac{m}{4\pi^2} \frac{ K_1( m \sqrt{ -x^2 + i \epsilon } ) }{ \sqrt{ -x^2 + i \epsilon } } \ , \end{align*} \tag{2.26}\] where \(K_1(x)\) is a modified Bessel function of the second kind. The \(\epsilon \to 0^+\) limit is a weak limit, it needs to be understood in the sense of distributions, i.e. it is needs to be calculated after the 2-point function has been integrated against a test function (as you would do in the definition of the Cauchy principal value of \(1/x\)). The Wightman 2-point function is instead given by \[ \begin{align*} \langle \Omega | \phi(x) \phi(0) | \Omega \rangle = & \int \frac{d^4 p}{(2\pi)^4} \, \theta(p_0) \, 2\pi \delta(p^2-m^2) \, e^{-i p x} \nonumber \\ = & \lim_{\epsilon \to 0^+} \frac{m}{4\pi^2} \frac{ K_1( m \sqrt{ -(x_0 - i \epsilon)^2 + \vec{x}^2 } ) }{ \sqrt{ -(x_0 - i \epsilon)^2 + \vec{x}^2 } } \ . \end{align*} \tag{2.27}\] Again, the \(\epsilon \to 0^+\) limit is a weak limit, it needs to be understood in the sense of distributions. Notice that the only difference between the time-ordered and the Wightman 2-point function is the way in which the singularity at \(x^2 = 0\) and the cut of the square root (which is hit for \(x^2>0\)) are approached from the complex plane.

On the other hand, by smearing distributions with Schwartz functions, one obtains smooth functions. This means that the vacuum expectation values of the smeared fields \(\phi^f(x)\), i.e. \[ \begin{gather*} \langle \Omega | \phi^{f_1}(x_1) \phi^{f_2}(x_2) \dots \phi^{f_n}(x_n) | \Omega \rangle = \int \left[ \prod_{k=1}^n d^4 x_k \, f_k(x_k-y_k) \right] \ \langle \Omega | \phi(y_1) \phi(y_2) \cdots \phi(y_n) | \Omega \rangle \ , \end{gather*} \tag{2.28}\] are smooth functions of the \(x_i\)’s.

2.5 Problems

Download Matthew Black’s solutions.

Problem 2.1 Prove that smeared fields transform under translations in the same way as local fields, i.e. \[ \begin{gather*} \phi^f (x + a) = e^{i P a} \phi^f(x) e^{-i P a} \ . \end{gather*} \] What happens under Lorentz transformations?

Problem 2.2 Define the Fourier transform of a field operator \(\phi(x)\) as \[ \begin{gather*} \tilde{\phi}(p) = \int d^4 x \, e^{i p x} \, \phi(x) \ . \end{gather*} \] Let \(| p \rangle\) be an eigenstate of the four-momentum operator \(P_\mu\) with eigenvalue \(p_\mu\). Prove that \(\tilde{\phi}(q) | p \rangle\) and \(\tilde{\phi}(q)^\dag | p \rangle\) are eigenstates of \(P_\mu\) and calculate the corresponding eigenvalues. Argue that \(\tilde{\phi}(q)^\dag\) and \(\tilde{\phi}(q)\) act as creation and annihilation operators for the four-momentum.

Problem 2.3 Consider two observables \(A\) and \(B\) which, for simplicity, are assumed to have only discrete non-degenerate eigenvalues. Denote by \(a_n\) and \(b_m\) the eigenvalues of \(A\) and \(B\), respectively, and by \(| a_n \rangle\) and \(| b_m \rangle\) the corresponding eigenstates. Consider a generic normalized state \(|\psi\rangle\). We imagine two different measurement protocols:

  • In the first protocol, we measure \(A\) on the state \(|\psi\rangle\) first, and then \(B\).
  • In the second protocol, we measure \(B\) on the state \(|\psi\rangle\) first, and then \(A\).

Recall that the state changes as a consequence of the measurement procedure. Let \(p_{n,m}\) (resp. \(q_{n,m}\)) be the probability of obtaining \(a_n\) as the result of the measurement of \(A\) and \(b_m\) as the result of the measurement of \(B\) in the first (resp. second) protocol. Write a formula for \(p_{n,m}\) and \(q_{n,m}\) in terms of the eigenstates \(| a_n \rangle\) and \(| b_m \rangle\) and the state \(|\psi\rangle\). Show that the order in which the observables are measured does not matter (for any state \(|\psi\rangle\)) if and only if the two observables commute.

Problem 2.4 In the free scalar case, prove the formulae: \[ \begin{align*} & \langle \Omega | \text{T} \{ \phi(x) \phi(0) \} | \Omega \rangle = \lim_{\epsilon \to 0^+} \int \frac{d^4 p}{(2\pi)^4} \, \frac{i \, e^{-i p x}}{p^2 - m^2 + i \epsilon} \ , \\ & \langle \Omega | \phi(x) \phi(0) | \Omega \rangle = \int \frac{d^4 p}{(2\pi)^4} \, \theta(p_0) \, 2\pi \delta(p^2-m^2) \, e^{-i p x} \ . \end{align*} \]

Problem 2.5 In operatorial formalism and temporal gauge (\(A_0=0\)), QCD can be described in terms of the following fundamental fields in the Schrödinger picture:

  • the gluon field \(A_k(\vec{x}) = \sum_a A_k^a(\vec{x}) T^a\), where \(k=1,2,3\) is the spatial index, \(a=1,\dots,8\) is the colour index and \(T^a\) are the generators of the gauge group \(SU(3)\) with the normalization \(\tr(T^a T^b) = \frac{1}{2} \delta^{ab}\); for instance, one can choose \(T^a = \lambda^a / 2\), where \(\lambda^a\) are the Gell-Mann matrices;
  • the chromoelectric field \(E_k(\vec{x}) = \sum_a E_k^a(\vec{x}) T^a\), with the same conventions as for the gluon field;
  • the quark fields \(\psi_{\alpha i f}(\vec{x})\), where \(\alpha=1,2,3,4\) is the Dirac spinor index, \(i=1,2,3\) is the colour index and \(f=u,d\) is the flavour index. The fundamental fields satisfy canonical equal-time (anti)commutation relations: \[ \begin{align*} & [A_j^a(\vec{x}), E_k^b(\vec{y})] = i \delta_{jk} \delta^{ab} \delta^3(\vec{x}-\vec{y}) \ , \\ & \{ \psi_f(\vec{x}), \psi_g^\dag(\vec{y}) \} = I_{12 \times 12} \delta_{fg} \delta^3(\vec{x}-\vec{y}) \ . \end{align*} \]

Consider the operator \(G(\vec{x}) = \sum_a G^a(\vec{x}) T^a\) \[ \begin{gather*} G(\vec{x}) = D_k E_k(\vec{x}) + \sum_f \psi^\dag_f(\vec{x}) T^a \psi_f(\vec{x}) \ , \end{gather*} \] and its smeared version \[ \begin{gather*} G(\omega) = 2 \int d^3 x \, \tr [ \omega(\vec{x}) G(\vec{x}) ] \ , \end{gather*} \] where \(\omega(\vec{x}) = \sum_a \omega^a(\vec{x}) T^a\) is a test function which is smooth and vanishes at infinity. Notice that we smear here only in space, not in time.

Calculate the commutators of \(G(\omega)\) with the fundamental fields \(A_k^a(\vec{x})\), \(E_k^a(\vec{x})\), and \(\psi_f(\vec{x})\). Argue that \(G(\vec{x})\) can be interpreted as the generator of time-independent gauge transformations and that physical states must satisfy the condition \(G(\vec{x}) | \Psi \rangle = 0\).

Problem 2.6 With the notation of the previous problem, the Hamiltonian of QCD in temporal gauge (\(A_0=0\)) reads: \[ \begin{gather*} H = \int d^3 x \, \left\{ g^2 \tr E_k^2(\vec{x}) + \frac{1}{2g^2} \tr F_{jk}^2(\vec{x}) + \sum_f \bar{\psi}_f (- i \gamma_k D_k - m_f) \psi_f(\vec{x}) + \epsilon_0 \right\} \ , \end{gather*} \] with the following definitions: \[ \begin{align*} & F_{jk} = \partial_j A_k - \partial_k A_j + i [ A_j, A_k ] \ , \\ & D_k = \partial_k + i A_k \ , \\ & \bar{\psi}_f = \psi_f^\dag \gamma^0 \ . \end{align*} \] The additive constant \(\epsilon_0\) is chosen in such a way that the vacuum has zero energy.

Show that \([G(\vec{x}), H] = 0\), i.e. the Hamiltonian is invariant under time-independent gauge transformations.

Problem 2.7 With the notation of the previous two problems, consider the operators \[ \begin{align*} & Q_f = \int d^3 x \, \psi_f^\dag(\vec{x}) \psi_f(\vec{x}) \ , \\ & U_f(\alpha) = e^{i \alpha Q_f} \ , \end{align*} \]

  1. Calculate the commutator of \(Q_f\) with the fundamental fields \(A_k^a(\vec{x})\), \(E_k^a(\vec{x})\), \(\psi_g(\vec{x})\) and \(\psi_g^\dag(\vec{x})\).

  2. Calculate \(U_f(\alpha) A_k^a(\vec{x}) U_f^\dag(\alpha)\), \(U_f(\alpha) E_k^a(\vec{x}) U_f^\dag(\alpha)\), \(U_f(\alpha) \psi_g(\vec{x}) U_f^\dag(\alpha)\) and \(U_f(\alpha) \psi_g^\dag(\vec{x}) U_f^\dag(\alpha)\).

  3. If \(P\) is a generic product of the fundamental fields and their derivatives, relate \([Q_f , P]\) to the number of quark fields of flavour \(f\) appearing in the operator \(P\). The operators \(Q_u\) and \(Q_d\) are called up-quark number and down-quark number, respectively.

  4. Show that the Hamiltonian commutes with \(Q_f\). Notice that this implies that \(U_f(\alpha)\) is a symmetry of the theory and the corresponding generator \(Q_f\) are conserved charges. We say that the up-quark number and the down-quark number are conserved in QCD.

  5. Write the baryon-number and electric-charge operators in terms of \(Q_u\) and \(Q_d\).