# Article

# Article

- Physics
- Theoretical physics
- Quantum anomalies

# Quantum anomalies

Article By:

**Jackiw, Roman **Department of Physics, Massachusetts Institute of Technology, Cambridge, Massachusetts.

Last reviewed:2014

DOI:https://doi.org/10.1036/1097-8542.757605

- Example from quantum mechanics
- Role in quantum field theory

- Role in other branches of physics
- Links to Primary Literature

- Additional Readings

**Phenomena that arise when a quantity that vanishes according to the dynamical rules of classical physics acquires a finite value when quantum rules are used.** For example, the classical Poisson bracket for some entities may vanish; yet the corresponding quantum commutator may be nonzero—this is a commutator anomaly. Alternatively, the flow of some material current may satisfy a continuity equation by virtue of the classical equations of motion, indicating conservative flow; but upon quantization the continuity equation may fail and the flow may no longer be conservative in the quantum theory—this is an anomalous divergence (of the current in question). Since the forms of Poisson brackets and quantum commutators as well as the occurrence of continuity equations for currents are related to symmetries and conservation laws of the theory, quantum anomalies serve to break some symmetries and destroy some conservation laws of classical models. This violation of symmetry is not driven by explicit symmetry-breaking terms in the dynamical equations—rather the quantization procedure itself violates the classical symmetry. The mathematical reason for this phenomenon is that classical dynamics, involving a finite number of degrees of freedom, usually leads to a quantum theory on an infinite-dimensional vector space (Hilbert space), and this “infinity” gives rise to novel effects. * See also: ***Canonical coordinates and transformations**; **Conservation laws (physics)**; **Equation of continuity**; **Hilbert space**; **Nonrelativistic quantum theory**; **Symmetry laws (physics)**

The content above is only an excerpt.

for your institution. Subscribe

To learn more about subscribing to AccessScience, or to request a no-risk trial of this award-winning scientific reference for your institution, fill in your information and a member of our Sales Team will contact you as soon as possible.

to your librarian. Recommend

Let your librarian know about the award-winning gateway to the most trustworthy and accurate scientific information.

## About AccessScience

#### AccessScience provides the most accurate and trustworthy scientific information available.

Recognized as an award-winning gateway to scientific knowledge, AccessScience is an amazing online resource that contains high-quality reference material written specifically for students. Its dedicated editorial team is led by Sagan Award winner John Rennie. Contributors include more than 9000 highly qualified scientists and 39 Nobel Prize winners.

**MORE THAN 8500** articles and Research Reviews covering all major scientific disciplines and encompassing the *McGraw-Hill Encyclopedia of Science & Technology* and *McGraw-Hill Yearbook of Science & Technology *

**115,000-PLUS** definitions from the *McGraw-Hill Dictionary of Scientific and Technical Terms *

**3000** biographies of notable scientific figures

**MORE THAN 17,000** downloadable images and animations illustrating key topics

**ENGAGING VIDEOS** highlighting the life and work of award-winning scientists

**SUGGESTIONS FOR FURTHER STUDY** and additional readings to guide students to deeper understanding and research

**LINKS TO CITABLE LITERATURE** help students expand their knowledge using primary sources of information