I'm doing an exercise where I'm trying to formulate the phonological rules for adapting English words into Japanese ones using feature theory. I came across a problem early on where En. /æ/ turns into Jp. /a/, but I have no way to formulate this rule because [tense] doesn't apply to these two vowels, and they both seem to share the same feature (+syl -con +son -high +low -round). I found a resource that said that [a] lacks the [back] specification entirely, but that still doesn't help me. How should I go about this?
The letter æ has a pretty standard phonetic value, but the letter a is quite ambiguous. The IPA standard is that [a] is a front open vowel (notice that [æ] doesn't fall in one of the quantal 'named' positions), but the letter is generally used for the single low (open) vowel in most languages. If a language has only one low vowel, it is usually written as a, whether it is back, central or front. If there are two low vowels in a language, you have to use different letters, so you will find [a ɑ], [æ ɑ] or [æ a]. Norwegian, for example, has [ɑ a] (using proximity to received IPA value as the standard), although one could substitute [æ] for [a]. American English has [æ ɑ] though the vowel that isn't [æ] is also not as low or back as Norwegian [ɑ]. So bear in mind that the letters people use are an approximation, and if you are concerned with phonetic values, you have to dig deeper to see what the formant values are.
The use of [tense] on low vowels is controversial, but I should point out that IPA [a] and [æ] can be distinguished on the basis of tenseness, with [æ] being the lax version (under the SPE definition of what "tense" means). If you cannot negotiate on the value of [tense], then you cannot solve the problem using the feature system that you've referred to (this is classic SPE feature theory).
Underspecification is one escape, which is somewhat outside that system, where you could leave Japanese a unspecified for back. You would need to spell out what the underspecified values are for vowels of English vs. Japanese. Japanese vowel are a proper subset of English vowels (if we ignore the actual phonetics of Japanese u, as is traditional), so the matter mostly reduces to noting that English has more vowel contrasts than Japanese.
Your proposed common features for the two letters omits the feature [back], which is clearly contrastive in English (hat and hot are different words, and the vowels only differ in backness). The standard account of æ→a presumes that "a" really represents a back vowel, perhaps better written as [ɑ], and the low front vowel of English becomes the low vowel of Japanese (the low vowel is always back, in Japanese).