Internal, Spacetime, Discrete, and Antiunitary Symmetries

The labels internal, spacetime, continuous, discrete, unitary, antiunitary, and orientation reversing answer different questions. Internal versus spacetime says whether the symmetry moves spacetime points. Continuous versus discrete describes the topology of the symmetry group. Unitary versus antiunitary describes how the symmetry acts on complex quantum amplitudes. Orientation reversal is a property of the spacetime map. These axes are logically distinct, with one structural constraint: continuity from the identity makes the entire identity component unitary, so any antiunitary elements lie in disconnected components.

That independence is the central diagnostic. A finite internal symmetry can be unitary. When parity is a symmetry, its ordinary implementation is unitary and orientation reversing; when time reversal is a symmetry of an ordinary positive-energy QFT, it is antiunitary. The discussion here concerns ordinary invertible symmetries of a fixed relativistic QFT; gauge redundancy, anomalies, conformal symmetry, and CPT proofs have separate owners.

Throughout, a symmetry includes its action on states, operators, sectors, parameters, and boundary conditions—not merely a formal substitution in a Lagrangian.

Required background. What Is a Symmetry of a QFT? supplies the distinction between a transformation of field symbols and a faithful action on complete physical data.

Helpful background. Lorentz Field Representations and Poincaré Particle Representations supplies the representation language for spacetime actions. CPT: Hypotheses, Content, and Limits states the hypotheses and proof of the CPT theorem; this page uses $C$, $P$, and $T$ only as individual symmetry operations.

Internal versus spacetime action

For a unitary symmetry represented by $U(g)$, a local-operator multiplet may transform as

$$U(g)\,\mathcal O_a(x)\,U(g)^{-1} =R(g)_a{}^b\,\mathcal O_b(g^{-1}x).$$

This formula displays two logically separate actions. The matrix $R(g)$ acts on operator labels, while $g^{-1}x$ acts on the spacetime argument.

An internal symmetry leaves every spacetime point fixed, so $g^{-1}x=x$. The global phase rotation of a complex scalar,

$$\phi(x)\longmapsto e^{i\alpha}\phi(x),$$

is internal. A spacetime symmetry moves the argument and transforms tensor or spinor indices accordingly. Translations, Lorentz transformations, parity, and time reversal are spacetime symmetries.

The distinction need not imply a direct-product group. Spacetime transformations can act on internal symmetry data, and extensions can mix the two. “Internal” identifies a trivial action on spacetime, not a promise that the full symmetry group factorizes.

Continuous versus discrete is a topological distinction

A symmetry group has a continuous sector when its identity component $G_0$ is nontrivial. Elements of $G_0$ are connected to the identity and admit infinitesimal generators under the usual regularity assumptions, even if the full group has additional disconnected components. The scalar $U(1)$ rotation above is connected and continuous. Its subgroup

$$\phi\longmapsto e^{2\pi i k/N}\phi, \qquad k=0,1,\ldots,N-1,$$

is the discrete internal group $\mathbb Z_N$. A purely discrete group has a trivial identity component; it need not be finite.

Beyond the unitary identity-component constraint, topology alone does not decide how a disconnected transformation acts on complex amplitudes. The $\mathbb Z_N$ action is unitary. Charge conjugation can also be a unitary discrete internal symmetry, whereas time reversal, when it is a symmetry, is a discrete antiunitary spacetime operation. Likewise, a transformation can lie in a disconnected component of a continuous Lie group without becoming antiunitary.

Unitary and antiunitary implementations

Wigner’s theorem permits a transition-probability-preserving transformation of rays to be implemented unitarily or antiunitarily. For state vectors,

$$\begin{aligned} U(c_1|\psi\rangle+c_2|\chi\rangle) &=c_1U|\psi\rangle+c_2U|\chi\rangle,\\ \Theta(c_1|\psi\rangle+c_2|\chi\rangle) &=c_1^*\Theta|\psi\rangle+c_2^*\Theta|\chi\rangle. \end{aligned}$$

Both preserve norms and transition probabilities, but their inner-product laws differ:

$$\begin{aligned} \langle U\psi|U\chi\rangle &=\langle\psi|\chi\rangle,\\ \langle\Theta\psi|\Theta\chi\rangle &=\langle\psi|\chi\rangle^*. \end{aligned}$$

Consequently an antiunitary operation conjugates scalar coefficients in its adjoint action,

$$\begin{aligned} \Theta(c\mathcal O)\Theta^{-1} &=c^*\Theta\mathcal O\Theta^{-1},\\ \Theta i\Theta^{-1}&=-i. \end{aligned}$$

This is what antiunitary means. It is not the same as writing a complex-conjugated field. For example, charge conjugation of a complex scalar sends $\phi$ to $\phi^\dagger$ but is represented unitarily in the ordinary scalar theory. The unitary/antiunitary alternative and its relation to discrete spacetime transformations are developed in Weinberg 1995, Vol. I, §§ 2.2 and 2.6, pp. 50–55 and 74–81.

The square of an antiunitary transformation is unitary. Its value is extra representation data: on a specified sector it may be $+1$ or the projective value $-1$ associated with a Kramers pair; on a full QFT Hilbert space it may instead equal fermion parity $(-1)^F$ or another internal transformation. Antiunitarity alone does not determine it. Moreover, rephasing $\Theta$ cannot change its square, because

$$(e^{i\beta}\Theta)^2 =e^{i\beta}e^{-i\beta}\Theta^2 =\Theta^2.$$

Projective relations and more general extensions are developed in Quantum Implementations, Projective Actions, and Central Extensions.

The following comparison applies the independent axes to the scalar conventions used on this page. Read each column separately: group topology does not determine the Hilbert-space type, and the internal/spacetime axis does not determine whether orientation is reversed. For the discrete rows, the displayed operator action, its square, and any intrinsic phases are part of the convention rather than consequences of the coordinate map alone.

Ordinary symmetry actions in the page's complex-scalar convention

Action	Internal or spacetime	Continuous or discrete	Quantum implementation	Coordinate and coefficient convention
Phase rotation U(1)	Internal: each spacetime point is fixed.	Continuous and connected to the identity.	Unitary.	φ(x) → eiαφ(x); complex coefficients are not conjugated.
Residual ℤN rotation	Internal: each spacetime point is fixed.	Discrete; finite in this example.	Unitary.	φ(x) → e2πik/Nφ(x); no orientation reversal.
Charge conjugation C	Internal in the scalar convention.	Discrete.	Unitary.	Cφ(x)C−1 = φ†(x); conjugating the field does not conjugate arbitrary scalar coefficients.
Parity P	Spacetime.	Discrete.	Unitary in the stated positive-energy convention.	(t, x) → (t, −x); spacetime orientation reverses, while time orientation does not.
Time reversal T	Spacetime.	Discrete.	Antiunitary.	(t, x) → (−t, x) and T i T−1 = −i; spacetime and time orientation reverse.

These rows are examples, not implications among the column headings. In particular, charge conjugation shows that mapping $\phi$ to $\phi^\dagger$ can be unitary, while the residual $\mathbb Z_N$ shows that a discrete symmetry can be unitary. A different QFT or sector may change intrinsic phases and squares; antiunitarity itself fixes conjugate-linearity, not $\Theta^2$.

Why ordinary time reversal is antiunitary

Let $\Theta$ reverse the orientation of time. It must conjugate a time translation into its inverse:

$$\Theta U(a)\Theta^{-1}=U(-a), \qquad U(a)=e^{-iHa}.$$

If $\Theta$ were unitary, the left side would be $e^{-i(\Theta H\Theta^{-1})a}$, so the relation would require $\Theta H\Theta^{-1}=-H$. That is incompatible with a nontrivial QFT spectrum bounded below and unbounded above. If $\Theta$ is antiunitary, it also sends $i$ to $-i$, and instead

$$\Theta e^{-iHa}\Theta^{-1} =e^{+i(\Theta H\Theta^{-1})a}.$$

The required inverse time translation is then obtained with $\Theta H\Theta^{-1}=H$. Antiunitarity preserves positive energy while reversing the time parameter. This positive-energy argument is the operational reason for the difference between parity and time reversal; see Weinberg 1995, Vol. I, § 2.6, pp. 74–81.

In four-dimensional Minkowski spacetime, the standard transformations act on coordinates as

$$\begin{aligned} \mathsf P:&\quad (t,\mathbf x)\longmapsto(t,-\mathbf x),\\ \mathsf T:&\quad (t,\mathbf x)\longmapsto(-t,\mathbf x). \end{aligned}$$

Each has four-dimensional Jacobian determinant $-1$, so each reverses spacetime orientation, but only $\mathsf T$ reverses time orientation. Their standard actions on the Hamiltonian $H$, momentum $\mathbf P$, and angular momentum $\mathbf J$ are

$$\begin{array}{c|ccc} &H&\mathbf P&\mathbf J\\ \hline \mathsf P\ \text{(unitary)}&H&-\mathbf P&+\mathbf J\\ \mathsf T\ \text{(antiunitary)}&H&-\mathbf P&-\mathbf J \end{array}$$

The different sign of $\mathbf J$ follows because angular momentum is an axial vector under parity but reverses with motion under time reversal. Concrete scalar and spinor realizations are worked in Schwartz 2014, §§ 11.4–11.6, pp. 193–200.

Euclidean reflection needs additional data

After Wick rotation, $t=-i\tau$, the coordinate map $\tau\mapsto-\tau$ is an ordinary reflection of a Euclidean background. By itself it is neither unitary nor antiunitary: those words describe a Hilbert-space implementation. To encode a Lorentzian antiunitary symmetry in Euclidean correlators, one must also specify the involution on fields and sources, complex conjugation, and the reflection-positivity structure used in reconstruction.

Orientation reversal also changes the sign of orientation-sensitive tensors and integrals. A pseudoscalar coupling or topological term may therefore have to change sign or be restricted to special values. Coordinate invariance of a Euclidean action alone is not a complete quantum symmetry test; the measure, background data, and possible anomaly must also be checked.

Threaded complex-scalar example

Consider a charge-one complex scalar with

$$\mathcal L_0 =\partial_\mu\phi^*\partial^\mu\phi -m^2\phi^*\phi -\frac{\lambda}{2}(\phi^*\phi)^2.$$

At fixed real $m^2$ and $\lambda$, several independent classifications are visible.

• The phase rotation $\phi\mapsto e^{i\alpha}\phi$ is continuous, internal, and unitary. Its $\mathbb Z_N$ subgroup is discrete, internal, and unitary.
• With the conventional phase choice, charge conjugation $\mathsf C\phi(x)\mathsf C^{-1}=\phi^\dagger(x)$ is discrete, internal, and unitary. It exchanges particles and antiparticles.
• Parity may be chosen as $\mathsf P\phi(t,\mathbf x)\mathsf P^{-1}=\phi(t,-\mathbf x)$. It is discrete, spacetime, orientation reversing, and unitary.
• Time reversal may be chosen as $\Theta\phi(t,\mathbf x)\Theta^{-1}=\phi(-t,\mathbf x)$. It is discrete, spacetime, time-orientation reversing, and antiunitary. For this spin-zero implementation, $\Theta^2=1$.

Now add a controlled breaking term with $N\geq2$,

$$\Delta\mathcal L =h\phi^N+h^*(\phi^*)^N.$$

For fixed nonzero $h$, the continuous $U(1)$ is reduced to its faithful $\mathbb Z_N$ subgroup. In the phase conventions just chosen, $\mathsf C$ exchanges the two monomials while $\Theta$ conjugates their coefficients; either operation separately is a symmetry when $h$ is real. Their antiunitary product $\mathsf C\Theta$ preserves the Hermitian pair for any $h$. But the phase of a single $h$ is not invariant under rephasing $\phi$ and simultaneously redefining the discrete operators. Genuine symmetry breaking is diagnosed only after all couplings and allowed redefinitions are specified.

For example, with several terms $h_q\phi^q+h_q^*(\phi^*)^q$, a generalized time reversal $\Theta_\eta\phi\Theta_\eta^{-1}=e^{i\eta}\phi$ requires

$$e^{iq\eta}h_q^*=h_q \qquad\text{for every }q.$$

One common $\eta$ need not solve all these equations, so relative phases can obstruct every time-reversal implementation of this form even when each term separately admits one. That conclusion does not apply to the basic $\mathsf C\Theta$ action above: for this restricted scalar potential it still preserves every Hermitian pair. Operators with derivatives, tensor indices, or additional intrinsic phases require their own compatibility test.

This example separates three operations that notation often conflates: $\mathsf C$ conjugates the field but is unitary, $\Theta$ need not conjugate the field but is antiunitary, and the residual $\mathbb Z_N$ is discrete without being either charge conjugation or time reversal.

Common confusions

Discrete does not mean antiunitary. Finite internal groups are ordinarily represented by unitary operators unless the symmetry definition includes an antilinear component.

Complex conjugation of a field does not diagnose antiunitarity. Antiunitarity is conjugate-linearity on complex coefficients and state amplitudes. A unitary operator can exchange $\phi$ and $\phi^\dagger$.

Orientation reversal does not determine the Hilbert-space type. Parity and time reversal both reverse spacetime orientation in four dimensions, yet parity is unitary and time reversal is antiunitary in a positive-energy theory.

The square of time reversal is not fixed by the coordinate map. The maps $t\mapsto-t$ can accompany inequivalent actions on spin and internal labels. Record $\Theta^2$ rather than inferring it.

Classical invariance is not the final quantum test. A regulator, measure, boundary condition, or anomaly can obstruct a transformation that preserves the displayed classical action.

Check your understanding

For the scalar theory above, classify the full $U(1)$ phase rotation, its $\mathbb Z_N$ subgroup, $\mathsf C$, $\mathsf P$, and $\Theta$ along all four axes: internal/spacetime, continuous/discrete, unitary/antiunitary, and orientation preserving/reversing.

Check

The $U(1)$ rotation is internal, continuous, unitary, and spacetime-orientation preserving. Its $\mathbb Z_N$ subgroup is internal, discrete, unitary, and orientation preserving. The chosen $\mathsf C$ is internal, discrete, unitary, and orientation preserving. In four dimensions, $\mathsf P$ is spacetime, discrete, unitary, and orientation reversing. The chosen $\Theta$ is spacetime, discrete, antiunitary, and both spacetime- and time-orientation reversing. Products inherit each property by composition: for example, $\mathsf C\Theta$ remains antiunitary because a unitary operation composed with an antiunitary one is antiunitary.

What to carry forward

Classify an ordinary symmetry by asking four separate questions: Does it move spacetime? Is it connected to the identity? Is its quantum implementation linear or conjugate-linear? Does its spacetime map reverse orientation or time orientation? Then record representation data such as its square, kernels, and action on sources.

The next canonical treatments divide the remaining work:

• Quantum Implementations, Projective Actions, and Central Extensions owns projective group laws, phases, and nontrivial squares.
• Multiplets, Invariants, and Selection Rules uses unitary representation data to constrain operators and amplitudes.
• Conformal Algebra and Generators develops conformal spacetime symmetry and its real forms.
• Ground, KMS, and State-Selection Criteria owns developed symmetry-selected states on curved backgrounds.
• CPT: Hypotheses, Content, and Limits supplies the theorem that this taxonomy deliberately does not prove.

References

• Schwartz, Matthew D. Quantum Field Theory and the Standard Model. Cambridge: Cambridge University Press, 2014. DOI
• Weinberg, Steven. The Quantum Theory of Fields. Volume I: Foundations. Cambridge: Cambridge University Press, 1995. DOI