Established physics · Lesson 26 of 34
Symmetry, conservation, and the small print
A continuous symmetry gives a conservation rule; distinguish exact gauge charge from useful approximate or model-dependent numbers.
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501 · why some sums must balance
The simple intuition
If the rules for an experiment are the same today as tomorrow, there is a conserved quantity associated with time-translation symmetry: energy. If the rules are the same after moving the apparatus, momentum is conserved. Rotational symmetry connects to angular momentum. This is the core idea behind Noether’s theorem.
A bookkeeper at every vertex
Suppose a W⁻ contribution connects a muon to a muon neutrino, then to an electron and electron antineutrino. Electric charge starts at −1 and ends at −1. Four-momentum also balances in the complete process. A Feynman diagram encodes these constraints in its vertices and lines.
Electric charge is especially firm
The surviving electromagnetic gauge symmetry makes electric charge conservation an exact rule within the Standard Model. The positron’s +1 balances the electron’s −1 when a neutral state produces a pair. A proposed diagram that violates charge is not rescued by drawing more internal lines.
Baryon number as a useful count
Assign baryon number +1/3 to a quark, −1/3 to an antiquark, and 0 to a lepton. A proton’s three quarks sum to +1. In ordinary observed reactions this accounting works extremely well. It is an accidental symmetry of the Standard Model’s renormalizable interactions, not an independent gauge charge that absolutely forbids all conceivable baryon violation.
Lepton number needs care
Assign +1 to a lepton and −1 to its antilepton for familiar processes. Muon decay starts with +1 and ends with electron +1, muon neutrino +1, and electron antineutrino −1: total +1. Individual family lepton numbers are not exact because neutrinos oscillate. Whether total lepton number is exact in nature depends on the neutrino mass mechanism; lepton-number-violating processes are actively tested.
Anomalies and nonperturbative exceptions
The Standard Model has subtle electroweak nonperturbative processes that violate B+L while preserving B−L in the minimal setting. This does not undermine the ordinary low-energy bookkeeping. It tells us to label a rule as exact, accidental, approximate, or dependent on model assumptions.
Symmetry is not the same as a visible mirror
A symmetry is a change of description or transformation that leaves the relevant laws unchanged. Some symmetries describe physical transformations such as moving an experiment; gauge transformations also contain descriptive redundancy. The specific mathematical symmetry tells you which conservation argument applies.
Try it on beta decay
At the quark level, d → u + W⁻ can lead to e⁻ + anti-νe. Charge balances at each vertex: −1/3 = +2/3 −1 at the quark end, and −1 = −1 +0 when W⁻ yields the lepton pair. Baryon number of the quark remains +1/3. Lepton number of the electron and antineutrino adds +1−1=0.
A useful conserved count can have a different status from an exact gauge charge.
Takeaway: Always ask what symmetry protects a conservation statement and under which conditions.
Check your understanding
If physical laws are unchanged by shifting the time of an experiment, what is connected by Noether’s theorem?
- Energy conservation
- Color naming
- Exactly three families
Answer and explanation
Energy conservation
Time-translation symmetry corresponds to conserved energy.
What is the electric charge of an electron–positron pair together?
- Zero
- −2
- +2
Answer and explanation
Zero
−1 and +1 add to zero.
What baryon number is assigned to one quark?
- +1/3
- +1
- 0
Answer and explanation
+1/3
Three quarks in an ordinary baryon total +1.
Are individual electron and muon lepton-family numbers exact in nature?
- No; neutrino oscillation mixes flavors
- Yes; every neutrino always stays one flavor
- Yes; W exchange forbids flavor changes
Answer and explanation
No; neutrino oscillation mixes flavors
Flavor oscillation means separate family labels are not exact conservation laws.
Is total lepton number known to be absolutely exact?
- No; its status depends on the neutrino mass mechanism
- Yes; it follows solely from electric charge
- No; muon decay violates it by two
Answer and explanation
No; its status depends on the neutrino mass mechanism
Majorana-type mass mechanisms can violate lepton number; it is experimentally investigated.
Which statement about B and L is best?
- Useful in ordinary reactions, with known theoretical qualifications
- They are identical to electric charge
- They forbid all possible processes at every energy
Answer and explanation
Useful in ordinary reactions, with known theoretical qualifications
Accidental symmetries and electroweak effects make their status subtler.
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