Can changing local phase scales change a particle’s motion?

Comparing arrows at different sites

Place four sites at the corners of a square. Give each site a complex amplitude, drawn as an arrow. Its squared length gives the probability of finding the particle there. Its angle is the phase. An isolated arrow has no observable absolute direction; interference compares phases.

Number the sites $i=0,1,2,3$ from left to top to right to bottom, matching A, B, C and D in the experiment. Write their amplitudes as $\psi_i=\rho_i e^{i\theta_i}$ , with dimensionless modulus $\rho_i$ . The link $U_{ij}=e^{ia_{ij}}$ transports a phase from site $j$ to site $i$ before comparison. All angles are in radians.

This transport is the essential step. Cities can set their clocks differently if the timetable converts local times correctly. Local phase scales also need a conversion rule.

\psi_i^*U_{ij}\psi_j=\rho_i\rho_j e^{i(\theta_j-\theta_i+a_{ij})}

This product combines the amplitudes with their comparison rule. Its real part can enter the hopping energy and its imaginary part can enter probability current.

Changing both ends and the link

Choose an independent angle $\alpha_i$ at each site. A new scale adds $\alpha_i$ to the site phase. The link must compensate for the difference between its endpoints.

Substitute the transformations into the product above. The phase from the left amplitude, the link and the right amplitude cancel exactly. A gauge invariant retains its value under this change of local scales.

\psi_i\longrightarrow e^{i\alpha_i}\psi_i,\qquad U_{ij}\longrightarrow e^{i\alpha_i}U_{ij}e^{-i\alpha_j}

The link direction runs from the right index to the left index. Reversing it takes the complex conjugate.

\theta_j-\theta_i+a_{ij}\longrightarrow\theta_j+\alpha_j-\theta_i-\alpha_i+a_{ij}+\alpha_i-\alpha_j

Every added angle cancels. The covariant phase difference is unchanged.

Source: David Tong

What remains after a complete circuit

Transport the phase around the square. At intermediate sites the changes of scale cancel. Returning to the start can still leave an extra phase. The closed product is a Wilson loop, which retains information that local choices cannot erase.

Take the indexed loop product $W=U_{01}U_{12}U_{23}U_{30}$ . These Abelian phases commute, so only their sum matters. In continuous electromagnetism the corresponding phase is the oriented magnetic flux times charge divided by the reduced Planck constant. Its sign depends on the charge and transport conventions.

On a tree without a closed path, choose local scales successively from a root and every link can be set to unity. With a loop, the last edge retains the total loop phase. A loop therefore distinguishes a change of description from a change of physical conditions.

W=e^{i\Phi},\qquad \Phi=a_{01}+a_{12}+a_{23}+a_{30}\pmod{2\pi}

The flux phase is dimensionless. Adding a full turn leaves the Wilson loop unchanged.

A hand-calculated four-site example. Angles are in radians; these are mathematical reference values.
Case	Link angles $(a_{01},a_{12},a_{23},a_{30})$	Loop phase $\Phi$
Uniform links	$(0.2,0.2,0.2,0.2)$	$0.8$
Rotate only the local scale at site 0 by $0.5$	$(0.7,0.2,0.2,-0.3)$	$0.8$
Physically change one link phase	$(0.4,0.2,0.2,0.2)$	$1.0$

Redundancy and physical symmetry

A gauge transformation describes the same physical state with different variables. A physical symmetry maps a state to another state while preserving the laws. Rotating an isolated apparatus in space, for example, usually gives a differently oriented state with the same energy.

Freedom to choose a local phase concerns redundancy. Charges and transformations that remain nonzero at a boundary require separate treatment; the word “redundancy” does not remove electric charge. The next chapter takes the comparison between sites into continuous space.

Try the reasoning in an experiment

Compare gauge changes with flux changes

Predict what a new phase scale at one site will do to the loop reading.
Hold physical flux fixed. Drag one site and change the gauge choice. Record the loop and covariant phase difference.
Change the flux and repeat. Explain which operations change the description and which change the physical conditions.

Exercises and explanations

What is the Wilson loop if the link angles sum to $2\pi$ ?

Read the explanation

The complex exponential returns to its start after a full turn, so $W=1$ . This loop alone cannot distinguish angle sums separated by full turns.

Someone rotates one site amplitude and leaves every link unchanged. How can you test their claim that this is a gauge transformation?

Read the explanation

Calculate the covariant phase differences on its two adjacent links. They generally change, and so may hopping energy or current. A gauge transformation changes the links as well.

Why can all three remaining links be set to unity after deleting one edge of the square?

Read the explanation

Fix the scale at one site, then determine successive scales along the three edges. With no loop, a site never receives two conflicting requirements. This is a discrete gauge fixing.

Sources and reproduced scope

Derivations, tables and the plot are authored for this course. The reading locations below separate source results from our teaching extensions. Numerical examples are model references or labeled synthetic observations. No experimental dataset or source figure is copied.

David Tong, Gauge Theory (2018 lecture notes)

Sections 2.1–2.1.3 (connections and Wilson loops), 2.7.3 (Higgs phase), 3.1.1–3.2.2 (two-dimensional anomaly and Ward identities), 4.2.1 (link endpoints) and 4.3.1–4.3.3 (lattice fermions). Printed pages 26–37, 105–108, 122–139, 207–210, 220–228.

Aneesh V. Manohar, Introduction to Effective Field Theories, arXiv:1804.05863v1 (2018)

Sections 3.5 and 4 (integrating out and power counting), 5.5–5.7 (matching), 5.10.1, PDF equations (5.51)–(5.60) (QCD running and operator mixing), and 6.2–6.3 (field redefinitions). The scalar-channel examples are derived for this course. The known-log boundary expansion checks truncation of the published running factors; it does not calculate finite hard matching.