Half-gallon sized canning jars are available, but they are not recommended for canning many foods. In fact, the only products recommended by the manufacturer to can include apple juice and grape juice ONLY in a boiling water bath canner.
There are no other research-tested processes for half-gallon jars. Boiling water processes for other foods for jars larger than those published with recipes (usually pints and/or quarts) cannot be extended by any formula to a larger jar.
Historical canning resources may reference the use of half-gallon jars. However, these are not currently accepted or endorsed by the USDA, Cooperative Extension, or U.S. canning jar manufacturers.